Local Mean Velocity Research Articles

Solution of the equations of rotational motion for a class of torque-free gyrostats

with the same cylinders fitted with end plates to simulate the case of infinite length. The cylinders were mounted sufficiently rigid to prevent oscillation. The results are presented in Fig. 2. In accordance with earlier findings the Strouhal number of the infinite cylinder is approximately 0.2, computed from the frequency peak protruding from the otherwise turbulent spectrum. In case of the finite cylinder with aspect ratio 4, a regular motion indicated by a distinct frequency in the spectral distribution with S « 0.45 was found at Rer> ~ 2.4 X 10s. Apparently the frequency peak tended to become weaker with decreasing Reynolds number and at RCD ~ 5 X 104 a distinct frequency could no longer be discerned. The spectra obtained with the cylinder of aspect ratio 8 showed a shedding frequency with S = 0.2 at subcritical Reynolds numbers with the indication to become stronger with ReD increasing. The evaluation of Roshko's wake Strouhal number/' Sw = fDw/Uw showed for all measurements Sw « 0.2. Naumann2 found that due to the three dimensionality of the flow the separation line is no longer straight and thus the circulation of a separating vortex is no longer constant, which leads to a break up. This seems to be confirmed by some surface patterns presented by Gould, Raymer, and Ponsford.3 The surface patterns which were observed during this work, however, exhibit a very straight separation line and at the same time there was no indication of vortex shedding. It appears therefore that some additional explanations may be necessary. For any blunt body part of the flow going over the top of the body is being dragged into the wake. The strength of this downwash obviously decreases with increasing Reynolds number and is restricted to a certain length of the cylinder from top downwards, i.e., its over-all influence decreases with increasing aspect ratio. One may consider the downwash flow as a kind of air curtain acting to some extent like a splitter plate. Depending on its over-all strength it dampens the feedback from both sides of the cylinder arid thus the shedding mechanism. Two tests were made to check on this model. In the first case the cylinder with end plate was provided with a slit in the base open to atmosphere, so that by action of the low base pressure a weak jet was blown into the wake. This largely damped the shedding which was clearly observed when the slit was blocked. In the second test the downwash in the near region of the finite cylinder was prevented by a small plate behind the cylinder between half height and top. The presence of this plate accounted for the occurrence of regular shedding. The main observations may be summarized as follows: 1) the pressure distribution around a blunt cylinder, being similar to that of the infinite cylinder has the same Ren — dependence; 2) the region of highly three-dimensional flow in the wake is restricted to the first 3—4 cylinder diameters; 3) turbulence levels in the near wake reach values of approximate 80% of the local mean velocity; 4) macro scales in the nonperiodic wake are almost an order of magnitude smaller than in the periodic, i.e., shedding case. Their mean values over the cylinder height increase linearly with downstream distance after the first 2 to 3 cylinder diameters; 5) the universal wake Strouhal number appears to be a constant for finite and infinite cylinders alike;

A Dynamic Model and Measurement Technique for Studying Induction Air Swirl in an Engine Cylinder

A dynamic engine model and measurement technique have been developed to study induction air swirl in the cylinder of a 4-stroke-cycle engine. Fabrication of the cylinder and piston crown of the model from transparent plastic permitted flow visualization. Local mean flow directions were determined qualitatively using a tuft grid. Paddle wheels were used to determine the gross swirl characteristics of the flow. A single sensor hot wire anemometer placed in three orientations was used to measure the three orthogonal components of local mean velocity. An error analysis indicated that the technique gives results accurate within ±20 percent. The methods developed allow measurements under actual dynamic conditions of the gross swirl characteristics of different flows, as well as the detailed velocity fields at any time during the intake stroke. Sample results are presented to illustrate the application of the methods.

Wavetrains in inhomogeneous moving media

When a slowly varying wavetrain of small amplitude propagates in a general medium, changes of frequency and wavenumber are determined along definite paths known as rays. It is shown that, for a wide class of conservative systems in fluid dynamics changes in ampli­tude along the rays may be computed from conservation of wave action, which is defined as the wave energy divided by the intrinsic frequency. The intrinsic frequency is the frequency which would be measured by an observer moving with the local mean velocity of the medium. This result is the analogue for continuous systems of the adiabatic invariant for a classical simple harmonic oscillator. If the medium is time dependent or moving with a nonuniform mean velocity the intrinsic frequency is not normally constant, and wave energy is not conserved. Special cases include surface waves on a vertically uniform current in water of finite depth, internal gravity waves in a shear flow at large Richardson number, Alfvèn waves, sound waves, and inertial waves in a homogeneous rotating liquid in geostrophic mean motion.

Kinetics of sorption and volume change in clay-colloid pastes

The present paper analyses one-dimensional absorption (swelling) and desorption (shrinkage or consolidation) in two-component pastes of clay-colloid and electrolyte solution. The work deals specifically with the case where solution exchange with the paste column is possible only through its base. The extension to systems with 'top drainage' and 'double drainage' is trivial. The analysis combines Darcy's law applied to the flow of solution relative to the colloid particles with the continuity requirement. The hydraulic conductivity K and the moisture potential (i.e. minus the 'swelling pressure') � are taken to be arbitrary known functions of the volumetric solution content �. The fundamental flow equation which follows is basically a nonlinear partial integrodifferential equation. It can be restated as a Fokker-Planck equation with �-dependent 'diffusivity' and the 'drift velocity' spatially constant but unknown and time dependent. The dependent variable is conveniently taken as �, although it can be made � if this is desired. The specific problem is that of sorption consequent on a step-function change in applied load (i.e. of the value of � (or �) at the base of the column). The column is taken to be effectively semi-infinite, so that the solution applies only until swelling or shrinkage at the upper surface of the column first becomes significant. Under these conditions, the solution is of the similarity form x(�, t) = �(�)t1/2 where x is the space coordinate, t is time, and � is a function of � which is found by the solution of a nonlinear, ordinary, integrodifferential equation. With �(�) known, it is an elementary matter to deduce a great variety of properties of the sorption process: the instantaneous distribution of colloid content; the instantaneous rate of total volume change; the cumulative total volume change; the instantaneous local velocity and volume flux density of colloid; the instantaneous local mean velocity and volume flux density of solution relative to colloid; the instantaneous local absolute mean velocity and volume flux density of solution; the instantaneous local volume flux density of solution due to mass flow; and the displacement history of colloidal particles initially at various positions in the column.

Statistical-Mechanical Derivation of the Partial Molecular Stress Tensors in Isothermal Multicomponent Systems

Commencing with the general equations of molecular dynamics as used by Bearman and Kirkwood, the hydrodynamic equations of motion are derived for each component in an isothermal multicomponent system consisting of sensibly spherical nonreacting molecules. It was the primary purpose of the investigation of which this is a report to attempt to obtain a microscopic statistical basis for the macroscopic phenomenologic development of Snell and Spangler, especially in regard to their new formulation for the partial molecular stress tensors. An expansion of the number densities in pair space of perturbed nonequilibrium states alternate to that used by Bearman and Kirkwood is employed. Our expansion esplicitly recognizes the position vector arguments of both the number densities in singlet space and the local mean velocities of the individual components, as well as for the equilibrium pair correlation function. The quantities are then expanded about the central position vector in a Taylor's series. The results are not only corroborative of the previous phenomenologic development when the flows of the individual components are solenoidal and attention is confined to systems in which the kinetic contribution to the stress tensor is negligible in comparison to the contribution arising from intermolecular interaction, but they also provide a new microscopic theory of hydrodynamical flow in multicomponent systems.

Incompressible, turbulent boundary layers with arbitrary pressure gradients and divergent or convergent cross flows.

An effective viscosity hypothesis that has previously led to rather detailed predictions of equilibrium turbulent boundary layers is now applied to boundary layers with arbitrary mainstream pressure variations and with divergent or convergent cross flows. The empirical content of the hypothesis involving three empirical constants (one of which is the von Karman constant) is solely derived from constant pressure profile data. The present work is similar to the previous work in that the mean differential equations of motion are integrated numerically. This time, however, one must deal with partial differential equations instead of the ordinary differential equations applicable to equilibrium flows. An important result is that prediction of the skin-friction coefficient and separation is very good. For the data considered, it is apparent that the effective viscosity hypothesis is not restricted to equilibrium flows; a corollary is that the effective viscosity can be related to the local mean velocity profile to some reasonable but undetermined degree of approximation. A further result is that for a particular three-dimensional, divergent flow experiment the measured cross-flow profiles agree with those calculated using a scalar effective viscosity.

Effects of flow conditions on the efficiency of a sedimentation vessel

Removal efficiency in a sedimentation vessel is greatly subject to the motion of fluids. Ordinary rectangular sedimentation vessels are large in scale and are affected by various environmental conditions. Hence, the features of flow in such vessels are too involved to be strictly analysed based on hydrodynamic equations of fluid-motion. Consequently, no theoretical approaches to predict the removal efficiency in a practical sedimentation vessel are yet available, in spite of much previous work on this problem. In general, the properties of flow are characterized by two types of motion of fluids i.e. the one is that of local mean velocity and the other is homogeneous turbulence. In this paper, these types of motion are represented separately by simple flow models, named “diffusion model” and “circulation model”, respectively. These simple models can give a reasonable interpretation of the complicated motion of fluids in a sedimentation vessel, and the removal efficiency is also derived in a simple form. Moreover, the calculated values obtained by these models are found to be in good agreement with the experimental data.

Some Turbulence Measurements in Water

The results of limited turbulence measurements in water in a free-surface flow are presented. A commercially available hot-wire anemometer of the constant-temperature type was used, and the recording and reduction of data were accomplished using an IBM 7094-7040 digital computing system and its peripheral analog-to-digital conversion equipment. The local temporal mean velocities, root mean squares of the turbulent velocity fluctuations, turbulent energy spectra, and auto-correlation curves were obtained directly from the digital computer. The relative turbulent intensities ranged from 4% to 5% near the water surface to 16% near the channel bottom for two cases having R of 66,200 and 91,100. Energy spectra showed that most of the turbulent energy was concentrated at frequencies less than 2 cps to 4.5 cps. The macro-scale of the turbulence in the flow direction was on the order of 80% of the depth and the turbulent microscale in that direction was approximately 10% of the depth for the experiment at the lower Reynolds number.

Waves on the erodible bed of an open channel

In the first part of this work the stability of the erodible bed of a stream with a free surface is studied within the framework of classical hydraulics, in which the velocity variation with depth is reduced to a single mean velocity and the bed friction is related in a general way to the local depth and mean velocity. Only two-dimensional motions can be studied in this way. In considering bed friction and a difference in phase between deposition and the mean velocity gradient along the channel, this work combines aspects of earlier studies of Exner and Kennedy.In the absence of a phase difference between erosion and mean velocity, the analysis proceeds without linearization. When the development from an equili- brium flow down a uniform slope is considered, the bed waves formed under subcritical flows are found to move downstream, while those under super-critical flows upstream; in both régimes bed waves are damped. In each case the side of a wave facing in the direction of motion is the steeper.When a phase shift is introduced as well, the analysis is carried forward only after linearization. The primary effect of bed slope and friction is a shift in the ranges of phase angle for which growth can take place and a corresponding alteration in the wavelengths for maximum growth. Friction also reduces bed-wave celerity. Consideration is given to the physical processes represented by the artifice of a phase difference between erosion and mean velocity gradient.The second section of this investigation concerns two-dimensional potential flow over a wavy stream bed. This problem is considered from a point of view different from that adopted previously by Kennedy; a modified criterion is proposed for the maximum Froude number at which bed waves will form. It is in better agreement with measured data.In the third part of this paper, the potential analysis is extended to include class of three-dimensional motions. The conditions are found for the formation of dunes (bed waves 180° out of phase with the surface waves above) and anti-dunes (the two in phase). The criterion separating dunes and antidunes for two dimensions is found to give a lower limit on the Froude number for antidunes of the more general three-dimensional class. In the antidune regime the streamwise perturbation to the velocity can change sign between surface and stream bed; the limit for this is determined.The erosion equation relating changes in bed level to local stream speed is generalized to include sediment convection in two dimensions. With this equation, three distinct regimes of bed-wave motion are found: at low Froude numbers, dunes moving downstream; at higher Froude numbers, antidunes moving upstream; and, finally, antidunes moving downstream. The criterion separating the last two r6gimes is just that mentioned above as the limit of flows whose velocity perturbation has the same sign from surface to stream bed. It is argued that this fundamental change in the flow marks also the end of the region of growth of small bed waves. It is found that three-dimensional dunes and upstream-moving antidunes can exist beyond the two-dimensional limit, but that the latter applies for all bed waves such that the ratio of stream depth to wavelength (d/λ) is small. This explains why the modified criterion for two dimensions provides for small d/λ an envelope for data obtained from observations of a wide variety of bed forms, but fails to do so for larger d/λ. The celerity of bed waves beneath three-dimensional flows is discussed. It is suggested that a class of waves occurring in natural streams will move more slowly than would two-dimensional waves in similar conditions.The work concludes with a comparison of several methods of modelling erosive flows.

INTERACTION BETWEEN AN ADVANCING SHOCK WAVE AND OPPOSING JET FLOW

The interaction between a plane shock wave advancing in a shock tube and an opposing axisymmetric jet flow was recorded experimentally and analyzed. Shadowgraph, schlieren, and interferometric observations revealed that the shape of the shock distortion (bulge) caused by the jet flow closely resembled that of the local jet mean velocity distribution. Oblique shock relations, applied across shock bulge segments, yielded local flow jump parameters. The density distribution thus obtained compared favorably with interferometric results. Beginning at the bulge apex as a maximum the flow field exhibited several adjacent density maxima and minima in the axial direction. Within 1J and 10 nozzle diameters in the radial and axial direction, respectively, these oscillations attenuated, and the density approached the uniform value observed behind the undisturbed normal main shock front. Applying mass and momentum conservation relations and assuming isentropic processes in the axial direction, corresponding flow field quantities were obtained. An intense, intermittent compression wave system with wavelength nearly equal to the jet diameter appeared behind the moving shock front. The intensity of the propagating discrete wavelets was found to be of the order of 170 db and was observed to diminish in accordance with moving pointsource emission theory.

The characteristics of the turbulence in the mixing region of a round jet

Measurements in the mixing region of a 1 in. diameter cold air jet are described for Mach numbers ranging from 0.2 to 0.55. The statistical characteristics of the turbulence in the first few diameters of the flow may be expressed in terms of simple kinematic similarity relationships. These are based on the jet diameter and the distance downstream from the jet orifice as length-scales, and the inverse of the local shear as a time-scale. The experiments show that the integral time scale of the turbulence in a frame convected with the maximum energy of the turbulent motion is inversely proportional to the local shear.The most interesting result obtained is that the local intensity of the turbulence is equal to 0.2 times the shear velocity. This velocity is defined as the product of the local integral length-scale of the turbulence with the local shear. The local intensity is defined as the R.M.S. value of the local velocity fluctuations divided by the jet efflux velocity. It was found that the length-scale is proportional to the distance from the jet orifice, while the maximum shear is also related to this distance as well as to the jet efflux velocity. These two similarity relations break down close to the jet orifice and change beyond the first six or so diameters downstream. The convection velocity is not equal to the local mean velocity but varies slowly over the region of maximum shear when it is just over half the jet efflux velocity. The measurements of other observers fit the relationships obtained quite well. From these relationships it is possible to calculate the noise generated by the mixing region of a given jet directly, using expressions derived by Lilley (1958).

Characteristics of transition flow between parallel plates

AbstractMeasurements of pressure drop and mean local fluid velocities have been made in a smooth rectangular duct of large aspect ratio. Data have been taken on the steady, isothermal flow of water at room temperature in the viscous, transition, and lower turbulent ranges of flow. Impact probes were installed in the center of the stream, where flow between infinitely broad parallel plates was closely approximated. The limits of the transition range are discussed, and mean local fluid velocities are correlated. Comparison is made with transitional behavior in smooth tubes.

Transitional velocity patterns in a smooth concentric annulus

AbstractProfiles of mean local velocities have been experimentally determined in a smooth, concentric, horizontal annulus having a radius ratio of 0.331. The test fluid was water at room temperature flowing steadily at Reynolds numbers in the viscous, transition, and lower turbulent ranges. The transitional profiles, obtained by means of an impact probe, are summarized and discussed. Limits of the transition zone are established, and variation of the radius of maximum velocity with Reynolds number is confirmed.