Showing papers on "Turbulence published in 1970"

A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers

Abstract: The three-dimensional, primitive equations of motion have been integrated numerically in time for the case of turbulent, plane Poiseuille flow at very large Reynolds numbers. A total of 6720 uniform grid intervals were used, with sub-grid scale effects simulated with eddy coefficients proportional to the local velocity deformation. The agreement of calculated statistics against those measured by Laufer ranges from good to marginal. The eddy shapes are examined, and only the u-component, longitudinal eddies are found to be elongated in the downstream direction. However, the lateral v eddies have distinct downstream tilts. The turbulence energy balance is examined, including the separate effects of vertical diffusion of pressure and local kinetic energy.It is concluded that the numerical approach to the problem of turbulence at large Reynolds numbers is already profitable, with increased accuracy to be expected with modest increase of numerical resolution.

Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties

Abstract: Publisher Summary This chapter discusses heat transfer and skin friction in turbulent pipe flow with variable physical properties. The constant properties solution has been considered only so far as is necessary for the flow and heat transfer analysis with variable physical properties. The chapter highlights analytical methods to describe heat transfer mechanisms for constant liquid properties quite satisfactorily and to take into account the influence of the variation of physical properties with temperature versus heat transfer and skin friction in a number of important cases. Disagreement between theoretical and experimental results observed in other cases, in particular, with a considerable change in physical properties over the flow cross section, may be attributed to imperfect methods of estimating the effect of the variation of physical properties on turbulent diffusivity. Important experimental material has been accumulated on heat transfer and skin friction for variable physical properties. However, certain portions of this material possess relatively low accuracy that prevents its successful use. For a number of important cases, there has been no systematic data collection or that which is available is scanty and contradictory. Therefore, the need for further experimental investigations, with a high degree of accuracy, into the fluid mechanics and heat transfer for variable physical properties is quite urgent.

Transport Equations in Turbulence

Abstract: Turbulence transport equations, describing the dynamics of transient flow of an incompressible fluid in arbitrary geometry, have been derived in such a manner as to incorporate the principles of invariance (tensor and Galilean) and universality. The equations are described in detail and their applicability is demonstrated by comparison of solutions with experiments on turbulence distortion and on the turbulence in the flow between flat plates.

The mechanics of an organized wave in turbulent shear flow

Abstract: Some preliminary results on the behaviour of controlled wave disturbances introduced artificially into turbulent channel flow are reported. Weak plane-wave disturbances are introduced by vibrating ribbons near each wall. The amplitude and relative phase of the streamwise component of the induced wave is educed from a hot wire signal, allowing the wave speed and attenuation characteristics and the wave shape to be traced downstream. The normal component and wave Reynolds stress have been inferred from these data. It appears that Orr–Sommerfeld theories attempted to date are inadequate for description of these waves.

Aerodynamic sound generation by turbulent flow in the vicinity of a scattering half plane

Abstract: The presence of the edge of a half plane in a turbulent fluid results in a large increase in the noise generated by that fluid at low Mach numbers. The parameter which is important is the product then the farfield sound has the same features as would be predicted by geometrical acoustics. The edge does not produce any significant sound amplification.

The two-dimensional mixing region

Abstract: The two-dimensional incompressible mixing layer was investigated by using constant-temperature, linearized hot wire anemometers. The measurements were divided into three categories: (1) the conventional average measurements; (2) time-average measurements in the turbulent and the non-turbulent zones; (3) ensemble average measurements conditioned to a specific location of the interface. The turbulent energy balance was constructed twice, once using the conventional results and again using the turbulent zone results. Some differences emerged between the two sets of results. It appears that the mixing region can be divided into two regions, one on the high velocity side which resembles the outer part of a wake and the other on the low velocity side which resembles a jet. The binding turbulent–non-turbulent interfaces seem to move independently of each other. There is a strong connexion between the instantaneous location of the interface and the axial velocity profile. Indeed the well known exponential mean velocity profile never actually exists at any given instant. In spite of the complexity of the flow the simple concepts of eddy viscosity and eddy diffusivity appear to be valid within the turbulent zone.

An eddy cell model of mass transfer into the surface of a turbulent liquid

Abstract: Experimental gas absorption studies for bubbles transported in turbulent pipe flow of water strongly indicate that liquid phase controlled mass transfer is due to surface renewal by turbulent eddies. Predictions of transport behavior from the conditions of turbulent flow cannot be made in support of this mechanism because no satisfactory theory of turbulent transport near a gas-liquid interface is available. This work considers a model of the hydrodynamic behavior near the surface which provides a link between the observed mass transfer behavior and the state of the turbulent field. In this model, the very small scales of turbulent motion are considered to be controlling. These motions are idealized, and their flow and mass transfer behavior are solved analytically. The overall result for eddies of various sizes is related to the turbulent energy spectrum by using only the easily accessible parameter ϵ, the energy dissipation rate. This model gives quantitative agreement to within a factor of 2 for three widely different experimental situations including gas-liquid and liquid-solid interfaces. However, the predicted Reynolds number dependence is somewhat higher than the experimental result. The model attempts to clearly define the basic physical process at the interface. Therefore, it indicates the direction for further experimentation needed to clarify the basic relationship between the mass transfer rates in the liquid phase and the hydrodynamic behavior of the turbulent liquid.

Experiments on nearly homogeneous turbulent shear flow

Abstract: With a transverse array of channels of equal widths but differing resistances, we have generated an improved approximation to spatially homogeneous turbulent shear flow. The scales continue to grow with downstream distance, even in a region where the mean velocity gradient and one-point turbulence moments (component energies and shear stress) have attained essentially constant values. This implies asymptotic non-stationarity in the basic Eulerian frame convected with the mean flow, behaviour which seems to be inherent to homogeneous turbulent shear flow.Two-point velocity correlations with space separation and with space-time separation yield characteristic departures from isotropy, including clear ‘upstream–downstream’ unsymmetries which cannot be classified simply as axis tilting of ellipse-like iso-correlation contours.The high wave-number structure is roughly locally isotropic although the turbulence Reynolds number based on Taylor ‘microscale’ and r.m.s. turbulent velocity is only 130. Departures from isotropy in the turbulent velocity gradient moments are measurable.The approximation to homogeneity permits direct estimation of all components of the turbulent pressure/velocity-gradient tensor, which accounts for inter-component energy transfer and helps to regulate the turbulent shear stress. It is found that its principal axes are aligned with those of the Reynolds stress tensor. Finally, the Rotta (1951, 1962) linear hypothesis for intercomponent energy transfer rate is roughly confirmed.

Toward a turbulent constitutive relation

Abstract: In an attempt to explain the failure of the various pure homogeneous strain experiments to reach equilibrium (and consequently to support the contention of Townsend of an equilibrium structure of the Reynolds stress dependent only on geometry), the nature of the general Reynolds stress-mean velocity relation is examined. It is shown that if homogeneous flows become asymptotically independent of initial conditions, and if the Reynolds stress bearing structure can be characterized by a single time scale (i.e.–at sufficiently high Reynolds number) then these flows behave like classical non-linear viscoelastic media, with the Reynolds stress structure dependent on the (strain-rate) (time scale) product. Thus, the existence of an equilibrium structure implies the existence of an equilibrium time scale and a universal value of the product. The ideas permitting Reynolds stress and mean velocity to be related are applied to the dissipative structure in homogeneous flows, and it is found that in such flow the time scale never ceases to grow, so that these flows can never reach an equilibrium structure. With the aid of an ad-hoc assumption these flows are examined in some detail, and the results of experiments are predicted with considerable accuracy. It is suggested that (inhomogeneous) flows having an equilibrium time scale may, in the homogeneous limit, be expected to display a universal structure. The small departure from universality induced by the large eddies associated with inhomogeneity may be adequately predicted by this same ad-hoc model.

On the transition to turbulent convection. Part 1. The transition from two- to three-dimensional flow

Abstract: In a horizontal convecting layer several distinct transitions occur before the flow becomes turbulent. These are studied experimentally for several Prandtl numbers from 1 to 104. Cell size plan form, transitions in plan form, transition to time-dependence, as well as the heat flux, are measured for Rayleigh numbers from 103 to 105. The second transition, occurring at around 12 times the critical Rayleigh number, is one from steady two-dimensional rolls to a steady regular cellular pattern. There is associated with this a discrete change of slope of the heat flux curve, coinciding with the second transition observed by Malkus. Transitions to time-dependence will be discussed in part 2.

Turbulent boundary-layer wall-pressure fluctuations on smooth and rough walls

Abstract: Turbulent boundary-layer wall-pressure measurements were made with ‘pinhole’ microphones three times smaller (relative to a boundary-layer displacement thickness) than microphones used in earlier work. The improved high-frequency resolution permitted examination of the influence of high-frequency eddies on smooth-wall pressure statistics. It was found that the space-time decay rate is considerably higher than previously reported. Measurements of cross-spectral density made with 5 Hz bandwidth filters disclosed low phase speeds at low frequency and small separation. Measurements were repeated on rough walls and parallels were drawn from knowledge of a smooth-wall boundary-layer structure to propose a structure for a rough-wall boundary layer. The effect of independently varying roughness height and separation on the large and small-scale turbulence structure was deduced from the measurements. It was found that roughness separation affected the very large-scale structure, whereas the roughness height influenced the medium and very small-scale turbulence.

The formation of streamwise vorticity in turbulent flow

Abstract: Mean streamwise vorticity in turbulent flow is shown to arise both from mean flow skewing and from the inhomogeneity of anisotropic wall turbulence. The structure of the Reynolds stress tensor is examined in several flows where the latter mechanism predominates. On the basis of a simple model for the anisotropy, the direction of the secondary currents is deduced for the corner boundary layer, the salient edge flow, and in the non-uniform nominally two-dimensional boundary layer.

A model for inhomogeneous turbulent flow

Abstract: A set of model equations is given to describe the gross features of a statistically steady or 9slowly varying’ inhomogeneous field of turbulence and the mean velocity distribution. The equations are based on the idea that turbulence can be characterized by ‘densities’ which obey nonlinear diffusion equations. The diffusion equations contain terms to describe the convection by the mean flow, the amplification due to interaction with a mean velocity gradient, the dissipation due to the interaction of the turbulence with itself, and the dif­fusion also due to the self interaction. The equations are similar to a set proposed by Kolmo­gorov (1942). It is assumed that both an ‘energy density’ and a ‘vorticity density’ satisfy diffusion equations, and that the self diffusion is described by an eddy viscosity which is a function of the energy and vorticity densities; the eddy viscosity is also assumed to describe the diffu­sion of mean momentum by the turbulent fluctuations. It is shown that with simple and plausible assumptions about the nature of the interaction terms, the equations form a closed set. The appropriate boundary conditions at a solid wall and a turbulent interface, with and without entrainment, are discussed. It is shown that the dimensionless constants which appear in the equations can all be estimated by general arguments. The equations are then found to predict the von Karman constant in the law of the wall with reasonable accuracy. An analytical solution is given for Couette flow, and the result of a numerical study of plane Poiseuille flow is described. The equations are also applied to free turbulent flows. It is shown that the model equations completely determine the structure of the similarity solutions, with the rate of spread, for instance, determined by the solution of a nonlinear eigenvalue problem. Numerical solutions have been obtained for the two-dimensional wake and jet. The agreement with experiment is good. The solutions have a sharp interface between turbulent and non-turbulent regions and the mean velocity in the turbulent part varies linearly with distance from the interface. The equations are applied qualitatively to the accelerating boundary layer in flow towards a line sink, and the decelerating boundary layer with zero skin friction. In the latter case, the equations predict that the mean velocity should vary near the wall like the 5/3 power of the distance. It is shown that viscosity can be incorporated formally into the model equations and that a structure can be given to the interface between turbulent and non-turbulent parts of the flow.

Initial Instability of Fine Bed Sand

Abstract: The initial entrainment characteristics of fine bed sand are objectively defined in terms of the measured distributions of critical instantaneous bed shear stress associated with the observed movement of individual surface grains. Critical flow conditions are then predicted by equating the lower extremes in these characteristic critical shear stress distributions to the upper extremes in the distribution of instantaneous bed shear stress produced by the particular type of background flow under consideration. The method is applied to the case of two-dimensional channel flow over a flat bed and yields sufficiently consistent results to suggest an extension to Shields' curve for small grain Reynolds numbers. The detailed experimental observations clearly illustrate the important role played by bed region turbulence in the interaction process between the fluid and the mobile bed grains constituting the deformable boundary.

Deposition of Particles in Turbulent Flow on Channel or Pipe Walls

Abstract: This paper describes a method for predicting the deposition of particles entrained in turbulent flow. The method predicts the deposition of particles ranging from molecular size to ∼100µ in diamete...

Turbulent dynamo action at low magnetic Reynolds number

Abstract: The effect of turbulence on a magnetic field whose length-scale L is initially large compared with the scale l of the turbulence is considered. There are no external sources for the field, and in the absence of turbulence it decays by ohmic dissipation. It is assumed that the magnetic Reynolds number Rm = u0l/λ (where u0 is the root-mean-square velocity and λ the magnetic diffusivity) is small. It is shown that to lowest order in the small quantities l/L and Rm, isotropic turbulence has no effect on the large-scale field; but that turbulence that lacks reflexional symmetry is capable of amplifying Fourier components of the field on length scales of order Rm−2l and greater. In the case of turbulence whose statistical properties are invariant under rotation of the axes of reference, but not under reflexions in a point, it is shown that the magnetic energy density of a magnetic field which is initially a homogeneous random function of position with a particularly simple spectrum ultimately increases as t−½exp (α2t/2λ3) where α(= O(u02l)) is a certain linear functional of the spectrum tensor of the turbulence. An analogous result is obtained for an initially localized field.

Survey of literature on flow characteristics of a single turbulent jet impinging on a flat plate

Abstract: Survey of literature on flow characteristics of single turbulent jet impinging on flat plate

Entrainment and the structure of turbulent flow

Abstract: Although the entrainment of non-turbulent fluid into a turbulent flow occurs across sharply defined boundaries, its rate is not determined solely by the turbulent motion adjacent to the interface but depends on overall properties of the flow, in particular, on those that control the energy balance. In the first place, attention is directed to the many observations which show that the motion in many turbulent shear flows has a structure closely resembling that produced by a rapid, finite, plane shearing of initially isotropic turbulence. The basic reasons for the similarity are the stability and permanence of turbulent eddies and the finite distortions undergone by fluid parcels in free turbulent flows. Next, the existence of eddy similarity and the condition of overall balance of energy are used to account for the variation of entrainment rates within groups of broadly similar flows, in particular mixing layers between streams of different velocities and wall jets on curved surfaces. For some flows which satisfy the ordinary conditions for self-preserving development, no entrainment rate is consistent with the energy balance and self-preserving development is not possible. Examples are the axisymmetric, small-deficit wake and the distorted wake. Finally, the implications of an entrainment rate controlled by the general motion are discussed. It is concluded that the relatively rapid entrainment in a plane wake depends on an active instability of the interface, not present in a constant-pressure boundary layer whose slow rate of entrainment is from ‘passive’ distortion of the bounding surface by eddies of the main turbulent motion. Available observations tend to support this conclusion.

Structure functions of turbulence in the atmospheric boundary layer over the ocean

Abstract: Structure functions of turbulent velocity fluctuations up to fourth order have been measured at several heights in the atmospheric boundary layer over the open ocean, and the results are compared with theoretical predictions for separations in the inertial subrange. The behaviour of second- and third-order quantities shows substantial agreement with the predictions of Kolmogorov's original theory over a wide range of separations, but the results of a recent modification of the theory, attempting to account for intermittency in the local dissipation rate, are also consistent with the data over somewhat shorter separation intervals. The behaviour of the measured fourth-order structure function disagrees with that predicted from Kolmogorov's original work, but good agreement is found with the results of the modified theory.

Vertical Transfer in Open Channel Flow

Abstract: Vertical turbulent mass transfer coefficients for suspended sediment particles and dye, and fall velocities of the sediment particles were determined experimentally in a large laboratory flume. Local values of the turbulent mass transfer coefficient, and turbulent fall velocities for the sand, were determined from an integrated form of the transfer equation. The results of the experiments indicate that: (1) Magnitude and distribution of the vertical turbulent mass transfer coefficient for dye and fine sediment are close to those of the momentum transfer coefficient, which lends added support to Reynolds' analogy for the equivalence of mass and momentum transfer in turbulent shear flow; (2) in comparison with dye, the mass transfer coefficient for medium sand is somewhat less in magnitude and its distribution indicates that the transfer coefficient is larger near the bed; and (3) the fall velocity of sediment particles is larger in turbulent channel flow than in quiescent water.

Turbulence and Flocculation

Abstract: Designers of flocculators are required to select the residence time, the number of flocculation compartments, the stirrer characteristics and the energy requirements, such that these function in an optimum manner in the treatment of a particular water. This paper presents a rational performance equation incorporating these parameters and demonstrates its validity with measurements employing a continuously operating model flocculation apparatus. Performance is shown to be determined by both the energy dissipation rate and the type of stirring equipment, but excessive energies result in floc breakup and reduced performance. At any particular performance a minimum residence time is shown to exist corresponding to an optimum energy dissipation. Anemometric measurements demonstrate a linear relationship between the mean square fluctuating velocity and the root mean velocity gradient computed from energy measurements. Whereas different stirrers have similar turbulence spectra, they display quite different performance coefficients.

Bounds for turbulent shear flow

Abstract: Bounds on the transport of momentum in turbulent shear flow are derived by variational methods. In particular, variational problems for the turbulent regimes of plane Couette flow, channel flow, and pipe flow are considered. The Euler equations resemble the basic Navier–Stokes equations of motion in many respects and may serve as model equations for turbulence. Moreover, the comparison of the upper bound with the experimental values of turbulent momentum transport shows a rather close similarity. The same fact holds with respect to other properties when the observed turbulent flow is compared with the structure of the extremalizing solution of the variational problem. It is suggested that the instability of the sublayer adjacent to the walls is responsible for the tendency of the physically realized turbulent flow to approach the properties of the extremalizing vector field.

Asymptotic theory of turbulent shear flows

Abstract: A theory is proposed in this paper to describe the behaviour of a class of turbulent shear flows as the Reynolds number approaches infinity. A detailed analysis is given for simple representative members of this class, such as fully developed channel and pipe flows and two-dimensional turbulent boundary layers. The theory considers an underdetermined system of equations and depends critically on the idea that these flows consist of two rather different types of regions. The method of matched asymptotic expansions is employed together with asymptotic hypotheses describing the order of various terms in the equations of mean motion and turbulent kinetic energy. As these hypotheses are not closure hypotheses, they do not impose any functional relationship between quantities determined by the mean velocity field and those determined by the Reynolds stress field. The theory leads to asymptotic laws corresponding to the law of the wall, the logarithmic law, the velocity defect law, and the law of the wake.

Turbulent flow drag reduction and degradation with dilute polymer solutions

Abstract: Experimental studies of drag reduction and polymer degradation in turbulent pipe flow with dilute water solutions of unfractionated polyethylene oxide are described. Drag reduction results indicate that the magnitude of the reduction cannot be correlated on the basis of weight average molecular weight, rather the phenomenon depends strongly on the concentration of the highest molecular weight species present in the molecular weight distribution. Polymer degradation in turbulent flow is found to be severe for high molecular weight polymers causing appreciable changes in drag reduction and molecular weight with the duration of flow. Data indicates that drag reduction exists in the limit of infinite dilution suggesting that the phenomenon is due to the interaction of individual polymer molecules with the surrounding solvent and that the extent of reduction is relatively independent of pipe diameter when a comparison is carried out at equal solvent wall shear stresses. Consideration of the high viscosity obtained with solutions in an irrotational laminar flow field suggests this is due to polymer molecule deformation and that this phenomenon is central to the mechanism of turbulent flow drag reduction.

Measurements of the Small-Scale Structure of Turbulence at Moderate Reynolds Numbers

Abstract: Measurements of probability densities and distributions, moments, and spectra obtained from the first two time derivatives of the streamwise velocity fluctuation in a mixing layer are presented. Probability distributions of squared derivatives are found to be nearly log normal. The skewness (S) and kurtosis (K) of the first derivative are compared with new atmospheric data at much higher Reynolds numbers. Reynolds-number dependence is indicated, in conflict with universal equilibrium theory. A modified theory using fluctuating dissipation predicts that S and K have power-law behavior with turbulent Reynolds number and are related by S ∝ K3/8, in general agreement with the trends of the limited data available.

Forced plume in a stratified fluid

Abstract: A turbulent buoyant jet operating in a linearly stratified fluid is investigated. By requiring similarity forms for the velocity, density difference, and shear stress and by adopting an integral representation of the governing equations, the rate of entrainment of ambient fluid into the plume is found to be an explicit function of the dependent variables. The entrainment is a function of the Reynolds stress, the form of the similarity profiles, and the local mean densimetric Froude number. The Reynolds stress term is shown to be related to the constant obtained if the entrainment rate is assumed to be proportional to the mean centerline velocity. A critical parameter of the analysis is the ratio of the form of the velocity profile to the form of the density difference profile. Values of this parameter near unity seem most appropriate. Numerical solutions are compared with experimental results to suggest values for the Reynolds stress term.

A three‐dimensional numerical investigation of the idealized planetary boundary layer

Abstract: The nonlinear equations of motion are integrated numerically in time for a region of x‐y‐z space of volume 3h × h × h, where h turns out to be a height slightly above the level where the wind first attains the geostrophic flow direction. Only the ideal case is treated of a horizontal lower boundary, neutral stability, horizontal homogeneity of all dependent mean variables except the mean pressure, and statistically steady state. The resulting flow patterns are turbulent and the eddies transport required amounts of momentum vertically. Topics which are investigated include the relative directions of stress, wind shear and wind; differences in Ekman wind spirals for the neutral numerical case and a stable atmospheric case; profiles of dimensionless turbulence statistics; effect of allowing the mean density to be either constant or to decrease with height; effect of the wind direction or latitude upon the turbulence intensities; and characteristic structure of the eddies in the planetary boundary layer.

Particle deposition from turbulent air flow

Abstract: A theoretical model was developed to describe effectively the turbulent deposition of particles through the boundary layer adjacent to a deposition surface. Particle transport to the deposition surface is described by effective particle eddy diffusivities. Two innovations from published deposition models are incorporated into the new model. The first innovation is the discarding of the concept that the particle velocity at the start of free-flight is simply related to the rms air motion directed toward the deposition surfaces. The second is that particle eddy diffusivities are not the same as eddy diffusivities for air momentum as is generally assumed. In this study, the particle eddy diffusivities are back-calculated from the deposition data. From these calculations it is concluded that particle turbulent diffusivities are greater than those for air momentum transfer in turbulent flow.

Scattering of aerodynamic noise by a semi-infinite compliant plate

Abstract: The acoustic scattering properties of a semi-infinite compliant plate immersed in turbulent flow are considered in the context of Lighthill's theory of aerodynamic noise The turbulent eddies are replaced by a volume distribution of quadrupoles, and the reciprocal theorem used to transform the quadrupole scattering problem into one of the diffraction of a plane acoustic wave This problem is solved by the Wiener–Hopf technique for the case when elastic forces in the plate are negligible, so that a local impedance condition relates the plate velocity to the pressure difference across the plate Strong scattering of the near-field into propagating sound occurs when certain types of quadrupole lie sufficiently close to the plate edge, and we derive explicit expressions for the scattered fields in various cases When fluid loading effects are small, and the plate relatively rigid, the results of Ffowcs Williams & Hall (1970) are recovered, in particular the U5 law for radiated intensity A quite different behaviour is found in the case of high fluid loading, when the plate appears to be relatively limp The radiated intensity then increases with flow velocity U according to a U6 law In aeronautical situations, surface compliance is negligible in its effect on the scattering process, and the U5 law must then apply provided the surface is sufficiently large On the other hand, the effect of appreciable surface compliance is to greatly inhibit the near-field scattering from the surface edge This weaker scattering is likely to be observed in underwater applications, where fluid loading effects are generally so high as to render unattainable the condition for a plate to be effectively rigid