Showing papers on "Turbulence published in 1969"
TL;DR: In this paper, the axisymmetric turbulent incompressible and isothermal jet was investigated by use of linearized constant-temperature hot-wire anemometers and the quantities measured include mean velocity, turbulence stresses, intermittency, skewness and flatness factors, correlations, scales, low-frequency spectra and convection velocity.
Abstract: The axisymmetric turbulent incompressible and isothermal jet was investigated by use of linearized constant-temperature hot-wire anemometers. It was established that the jet was truly self-preserving some 70 diameters downstream of the nozzle and most of the measurements were made in excess of this distance. The quantities measured include mean velocity, turbulence stresses, intermittency, skewness and flatness factors, correlations, scales, low-frequency spectra and convection velocity. The r.m.s. values of the various velocity fluctuations differ from those measured previously as a result of lack of self-preservation and insufficient frequency range in the instrumentation of the previous investigations. It appears that Taylor's hypothesis is not applicable to this flow, but the use of convection velocity of the appropriate scale for the transformation from temporal to spatial quantities appears appropriate. The energy balance was calculated from the various measured quantities and the result is quite different from the recent measurements of Sami (1967), which were obtained twenty diameters downstream from the nozzle. In light of these measurements some previous hypotheses about the turbulent structure and the transport phenomena are discussed. Some of the quantities were obtained by two or more different methods, and their relative merits and accuracy are assessed.
1,287 citations
TL;DR: In this article, the Kolmogorov-Prandtl turbulence energy hypothesis is formulated in a way which is valid for the laminar sublayer as well as the fully turbulent region of a one-dimensional flow.
1,018 citations
TL;DR: In this paper, it was shown that in spatially homogeneous two-dimensional turbulence, the mean square vorticity is unaffected by convection and can only decrease under the action of viscosity.
Abstract: Two‐dimensional and three‐dimensional turbulence have different properties, but both contain the two basic ingredients of randomness and convective nonlinearity, and some of the statistical hypotheses which have been proposed for three‐dimensional turbulence should be applicable to two‐dimensional motion. This justifies a numerical integration of the unaveraged equations of motion in two dimensions with random initial conditions as a means of testing the soundness of ideas such as those leading to the Kolmogoroff equilibrium theory. In spatially homogeneous two‐dimensional turbulence, the mean‐square vorticity is unaffected by convection and can only decrease under the action of viscosity. Consequently the rate of dissipation of energy tends to zero with the viscosity (v). On the other hand, the mean‐square vorticity gradient is increased by convective mixing, and it seems likely that the rate of decrease of mean‐square vorticity tends to a nonzero limit κ as ν → 0. This suggests the existence of a “casca...
901 citations
TL;DR: In this paper, the authors investigate for turbulent flow the fluid motions very near a solid boundary, and to create a physical picture which relates these motions to turbulence generation and transport processes.
Abstract: The objective of this study is to investigate for turbulent flow the fluid motions very near a solid boundary, and to create a physical picture which relates these motions to turbulence generation and transport processes. An experimental technique was developed which permitted detailed observations of the regions very near a pipe wall, including the viscous sublayer, without requiring the introduction of any injection or measuring device into the flow. This technique involved suspending solid particles of colloidal size in a liquid, and photographing their motions with a high-speed motion picture camera moving with the flow. To provide greater detail, the field of view was magnified.Fluid motions were observed to change in character with distance from the wall. The sublayer was continuously disturbed by small-scale velocity fluctuations of low magnitude and periodically disturbed by fluid elements which penetrated into the region from positions further removed from the wall. From a thin region adjacent to the sublayer, fluid elements were periodically ejected outward toward the centreline. Often there was associated with these events a zone of high shear at the interface between the mean flow and the decelerated region that gave rise to the ejected element. When the ejected element entered this shear zone, it interacted with the mean flow and created intense, chaotic velocity fluctuations. These ejections and resulting fluctuations were the most important feature of the wall region, and are believed to be a factor in the generation and maintenance of turbulence.
700 citations
TL;DR: The sparing of fatty streaking in localized areas is reported, at which they suggest low wall shear rate (the product of velocity gradient and fluid viscosity) is experienced and shown that acute elevation of shear rates on the aortic wall causes endothelial damage and increased permeability to lipids.
Abstract: THE patchy distribution of fatty streaking and early atheroma has been associated with arterial blood mechanics. Mustard et al.1 have noted occurrence of atheroma at sites which are thought to experience particle (platelet) deposition as the result of local rapid flow fluctuations (turbulence) or eddies. Others have proposed platelet deposition in regions of flow separation2. Texon3 invoked damage due to Bernoulli-type suction forces in areas of locally increased blood velocity; but this is considered implausible because the forces are negligible in physiological conditions in comparison with normal variations of mean blood pressure. Mitchell and Schwartz4 reported the sparing of fatty streaking in localized areas, at which they suggest low wall shear rate (the product of velocity gradient and fluid viscosity) is experienced. Fry5,6 has shown that acute elevation of shear rate on the aortic wall causes endothelial damage and increased permeability to lipids. These theories assign to fluid mechanics a causative role in atherogenesis.
618 citations
TL;DR: In this paper, a formal algebraic analogy is drawn between meteorological parameters such as the Richardson number and the parameters describing the effect of rotation or streamline curvature on a turbulent flow.
Abstract: A formal algebraic analogy is drawn between meteorological parameters, such as the Richardson number, and the parameters describing the effect of rotation or streamline curvature on a turbulent flow. The analogy between the phenomena is a good first approximation. Semi-quantitative use of the analogy to apply meteorological data to curved shear layers shows that the effects of curvature on the apparent mixing length are appreciable if the shear-layer thickness exceeds roughly 1/300 of the radius of curvature; larger effects may occur in compressible flow. Application of the Monin-Oboukhov formula considerably improves the agreement between prediction and experiment in boundary layers on curved surfaces.
555 citations
TL;DR: In this article, a detailed experimental study of turbulent boundary-layer development over rough walls in both zero and adverse pressure gradients is described, where the skin friction was determined by pressure tapping the roughness elements and measuring their form drag.
Abstract: This paper describes a detailed experimental study of turbulent boundary-layer development over rough walls in both zero and adverse pressure gradients. In contrast to previous work on this problem the skin friction was determined by pressure tapping the roughness elements and measuring their form drag.Two wall roughness geometries were chosen each giving a different law of behaviour; they were selected on the basis of their reported behaviour in pipe flow experiments. One type gives a Clauser type roughness function which depends on a Reynolds number based on the shear velocity and on a length associated with the size of the roughness. The other type of roughness (typified by a smooth wall containing a pattern of narrow cavities) has been tested in pipes and it is shown here that these pipe results indicate that the corresponding roughness function does not depend on roughness scale but depends instead on the pipe diameter. In boundary-layer flow the first type of roughness gives a roughness function identical to pipe flow as given by Clauser and verified by Hama and Perry & Joubert. The emphasis of this work is on the second type of roughness in boundary-layer flow. No external length scale associated with the boundary layer that is analogous to pipe diameter has been found, except perhaps for the zero pressure gradient case. However, it has been found that results for both types of roughness correlate with a Reynolds number based on the wall shear velocity and on the distance below the crests of the elements from where the logarithmic distribution of velocity is measured. One important implication of this is that a zero pressure gradient boundary layer with a cavity type rough wall conforms to Rotta's condition of precise self preserving flow. Some other implications of this are also discussed.
548 citations
TL;DR: In this paper, skin friction and mean-velocity profiles have been made in fully developed flows in pipes and channels in the Reynolds number range 1000 < Re < 10000, and observations of hot-wire signals indicate rather remarkable differences between two-dimensional and axially symmetric flows and also make it difficult to give a precise definition of the term "fully developed turbulent flow".
Abstract: Measurements of skin friction and mean-velocity profiles have been made in fully developed flows in pipes and channels in the Reynolds number range 1000 < Re < 10000 These measurements, and observations of hot-wire signals, indicate rather remarkable differences between two-dimensional and axially symmetric flows and also make it difficult to give a precise definition of the term ‘fully developed turbulent flow’
425 citations
TL;DR: In this paper, the authors investigated the nearshore circulation of water on a plane beach produced by a wave train, normally incident on the beach, which has a longshore variation in wave height.
Abstract: The nearshore circulation of water on a plane beach produced by a wave train, normally incident on the beach, which has a longshore variation in wave height is investigated theoretically. The radiation stress arising from the excess flux of momentum due to the presence of the waves (M. S. Longuet-Higgins and R. W. Stewart, 1964) is found to provide driving terms for a steady flow pattern only inside the surf zone. A circulation pattern is thus produced by a longshore variation in the radiation stress in the surf zone. In shallow water, the radiation stress is proportional to the square of the wave height. The nearshore circulation is therefore directly related to longshore variation in breaker height, currents flowing seaward where the breaker height is low. When the inertial terms are included in the vorticity equation, an increase in the effective Reynolds number produces a narrowing, and consequently a strengthening, of the Seaward flow, which suggests an explanation for the existence of the strong, narrow currents known as rip currents.
324 citations
TL;DR: In this article, the Navier-Stokes equations are numerically integrated in order to test the validity of Kraichnan's predictions on the structure of two-dimensional turbulence, and numerical results are consistent with these predictions, although truncation and aliasing errors restrict the useful spectral range of the calculations to less than that desirable for a fully satisfactory test.
Abstract: The two‐dimensional incompressible Navier‐Stokes equations are numerically integrated in order to test the validity of Kraichnan's predictions on the structure of two‐dimensional turbulence. The numerically simulated turbulence is produced by an artificial forcing function, constructed from a randomly varying set of Fourier modes lying within a narrow spectral band near scalar wavenumber ke. For the case of turbulence maintained by a constant rate of energy injection, Kraichnan predicts the development of two power‐law energy spectra, a k−5/3 range for frequencies less than ke, and a k−3 range for those greater than ke. The numerical results are consistent with these predictions, although truncation and aliasing errors restrict the useful spectral range of the calculations to less than that desirable for a fully satisfactory test. The predictions and results are apparently relevant to several areas of geophysical fluid dynamics and to the correct interpretation of other numerical simulation experiments.
227 citations
TL;DR: In this paper, the convective flow in an infinite horizontal fluid layer rotating rigidly about a normal axis was investigated for the special case of infinite Prandtl number and free boundary conditions.
Abstract: The convective flow in an infinite horizontal fluid layer rotating rigidly about a normal axis is investigated for the special case of infinite Prandtl number and free boundary conditions. For slightly supercritical Rayleigh numbers the solutions of the non-linear steady-state equations are derived approximately by an amplitude expansion. A stability calculation shows that no stable steady-state convective flow exists if the Taylor number exceeds the critical value 2285.
TL;DR: In this article, an experimental investigation on turbulent natural convection boundary layers has been conducted with water on a vertical plate of constant heat flux, with the emphasis on the turbulent regime.
Abstract: An experimental investigation on turbulent natural convection boundary layers has been conducted with water on a vertical plate of constant heat flux. Local heat transfer data are presented for laminar, transition, and turbulent natural convection, with the emphasis on the turbulent regime. The data extend to a modified Rayleigh number of 1016 for a threefold range in Prandtl number. The results indicate that natural transition occurs in the range 1012 < Ra* < 1014 ; i.e., fully developed turbulent flow occurs by Ra* = 104 . This latter value can be as low as 2 × 1013 with the use of a trip rod. The physical structure of the turbulent boundary-layer flow was studied using the combined time-streak marker hydrogen bubble method. Temperature data and temperature corrected velocity data obtained by hot-film sensors are presented for Ra* values between 8.7 × 1013 and 7.1 × 1014 . For the range of variables investigated, the major conclusions are (a) the local heat transfer coefficient exhibits a slight decrease with length, (b) confirmation that the vortex street layer in the transition region decays into a longitudinal-vortex-type structure, and (c) the outer portion of the thermal and velocity fields can be approximated by power profiles that fit almost all the data available to date.
TL;DR: An analysis of the spatial resolution of the Kovasznay vorticity meter and arrays for measuring velocity derivatives in isotropic turbulence is presented in this article, where Pao's three-dimensional spectrum is used, allowing calculations of the measured values of the one-dimensional VOR spectrum and mean-square velocity derivatives to be compared with their predicted values.
Abstract: Analyses of the spatial resolution of the Kovasznay vorticity meter and of arrays for measuring velocity derivatives in isotropic turbulence are presented. Pao's three-dimensional spectrum is used, allowing calculations of the measured values of the one-dimensional vorticity spectrum and mean-square velocity derivatives to be compared with their predicted values. It is found that for accurate measurement the array size should be of the order of the Kolmogorov microscale of the turbulence field.
TL;DR: In this paper, the authors checked the two-dimensionalality of the flow by measuring mean velocity and turbulence level profiles at different spanwise positions, and the flow was found uniform over 75% of the span (wind-tunnel width) at the trailing edge at the last downstream station, where wake measurements were taken.
Abstract: First the two-dimensionality of the flow was checked by measuring mean velocity and turbulence level profiles at different spanwise positions, and the flow was found uniform over 75% of the span (wind-tunnel width) at the trailing edge and over 60% of the span at the last downstream station, x = 240 cm, where wake measurements were taken. No detectable periodic components were found in the wake at any downstream station as evidenced either by wave analyzer or by correlation measurements taken with two probes located symmetrically to the centerline and placed near the location of maximum shear stress in the wake. The characteristics of the turbulent boundary layer at the trailing edge (x = 0) were the following: conventional thickness (U/Um = 0.99) d = 5.50 cm, momentum thickness d = 0.58 cm, and shape parameter H = 5*/6 = 1.44. At the same location, the friction velocity was found to be u*/Um = 0.046. This value was obtained by using Clauser's logarithmic law.5 The wall slope method using hot-wire measurements with correction for proximity of the boundary due to Wills 4 yielded a value of u*/Ua = 0.037. The thickness of the trailing edge expressed nondimensionally was quite small, u*h/v = 3; in other words, much smaller than the viscous sublayer. All data is presented in dimensionless form using 00, the momentum thickness at the trailing edge and the undisturbed flow velocity Um as reference quantities. Symmetry of the wake was found to be excellent and representative
TL;DR: In this article, Van Driest's theory was extended to turbulent flow near a nonporous wall, which enables the theoretical calculation of velocity profiles to be performed for a wider range of mass transfer rates, and gives good agreement with experimental data.
Abstract: : Van Driest's theory, which provides a continuous velocity and shear distribution for turbulent flow near a nonporous wall, is extended to turbulent flow near a porous wall. The new, modified theory enables the theoretical calculation of velocity profiles to be performed for a wider range of mass- transfer rates, and it gives good agreement with experimental data.
TL;DR: In this article, it has been shown that the thermocline microstructure has a predominant vertical scale of a few decimeters, the sheet waves have wavelengths in the range 5-10 meters, and the turbulence occurs at a Reynolds number of 200-1000.
Abstract: Dye-tracer experiments in the seasonal oceanic theormocline have revealed a microstructure with a significant fluctuation of shear and density gradient within a vertical distance of 1 cm. It has proved possible to relate the occurrence of localized spots of turbulence with this structure and with internal waves traveling along especially (statically) stable ‘sheets’ within the thermocline. The thermocline microstructure has a predominant vertical scale of a few decimeters, the sheet waves have wavelengths in the range 5–10 meters, and the turbulence occurs at a Reynolds number in the range 200–1000. Similar features are found on a very much larger scale in the atmosphere, although the methods of investigation (mainly instrumented aircraft) used here are incapable of yielding the detailed information obtained by divers working in the thermocline. It seems useful, therefore, to treat the observations of microstructure, waves, and turbulence in the thermocline as a possible model for the corresponding features in the atmosphere. This leads to some predictions concerning the atmosphere microstructure and also to suggestions for the planning of experimental investigations. The available evidence from the atmosphere appears to support the predictions that (1) the critical gradient Richardson number for reverse transition occurs at about unity regardless of the Reynolds number of the turbulence, and (2) that most of the atmosphere (with the exception of clouds and the boundary layer) is in laminar flow.
TL;DR: In this paper, a theory for the rise of a plume in a horizontal wind is proposed in which it is assumed that, for some distance downwind of a high stack, the effects of atmospheric turbulence may be ignored in comparison with the effects generated by the plume.
Abstract: A theory for the rise of a plume in a horizontal wind is proposed in which it is assumed that, for some distance downwind of a high stack, the effects of atmospheric turbulence may be ignored in comparison with the effects of turbulence generated by the plume. The theory, an extension of the local similarity ideas used by Morton, Taylor, and Turner,1 has two empirical parameters which measure the rate that surrounding fluid is entrained into the plume. Laboratory measurements of buoyant plume motion in laminar unstratified cross flow are used to estimate the empirical parameters. Using this determination of the parameters in the theory, the trajectories of atmospheric plumes may be predicted. To make such a prediction, the observed wind velocity and temperature as functions of altitude, and flow conditions at the stack orifice, are used in numerically integrating the equations. The resulting trajectories are compared with photographs, made by Leavitt, et al.,2 of TVA, of plumes from 500 to 600 ft high sta...
TL;DR: Torrance, Orloff & Rockett as discussed by the authors made an analytical study of the natural convection induced in an enclosure by a small hot spot centrally located on the floor, and solved the equations of fluid flow in axisymmetric cylindrical coordinates with the Boussinesq approximation.
Abstract: An analytical study was made of the natural convection induced in an enclosure by a small hot spot centrally located on the floor. The enclosure was a circular cylinder, vertically oriented, with height equal to radius. A Prandtl number of 0.7 (air) was assumed; the Grashof number (Gr) was based on cylinder height and hot spot temperature. The equations of fluid flow in axisymmetric cylindrical co-ordinates were simplified with the Boussinesq approximation. The equations were solved numerically with a computationally stable, explicit method. The computation, starting from quiescent conditions, proceeded through the initial transient to the fully developed flow. Solutions were obtained for Gr from 4 × 104 to 4 × 1010. The theoretical flows are in excellent agreement with experimentally observed laminar flows (Gr [lsim ] 1.2 × 109) which are discussed in a companion paper, Torrance, Orloff & Rockett (1969). Turbulence was observed experimentally for Gr [gsim ] 1.2 × 109. When the theoretical calculations were extended to Gr = 4 × 1010, a periodic vortex shedding developed, suggestive of the onset of laminar instability. The theoretical results reveal a √Gr scaling for the initial flow transients and, at large Gr, the velocities and heat transfer rates.
01 Jan 1969
TL;DR: In this article, a study of noise generation in turbulent premixed flames, turbulent diffusion flames, and in liquid-spray combustion of hydrocarbon fuels is described, and the authors show that all these flames may be represented acoustically as an assembly of monopole sound sources in their combustion zones.
Abstract: A study of the mechanism of noise generation in turbulent premixed flames, turbulent diffusion flames and in liquid-spray combustion of hydrocarbon fuels is described. It is shown that all these flames may be represented acoustically as an assembly of monopole sound sources in their combustion zones. The radiated sound pressure is dependent upon the rate of change of the rate of increase of volume of the fuel and oxidant during combustion. The rate of volume increase is proportional to the rate of consumption of the fuel and oxidant in the flame. To measure this quantity, an optical technique has been employed that relies on observations of changes in the intensity of emission from the free radicals C2 generated in the reaction zones of these flames. For the premixed flames, good quantitative agreement is obtained between the radiated sound pressure calculated from these intensity measurements, using simple acoustic theory, and the values recorded simultaneously with a microphone. Similar agreement is obtained for diffusion flames, with the assumption made that the fuel and oxidant burn in stoichiometric proportions. Qualitative agreement is obtained for the liquid-fuel flames. The mean intensities of emission from premixed flames, burning under varied conditions of turbulence, were found to depend linearly upon the total flow rate of the combustible mixture and to be independent of the conditions of turbulence. This observation is compatible with the wrinkled laminar flame model of turbulent flame propagation.
TL;DR: In this article, a model is developed which describes the adjustment of the flow of a hydrostatically neutral fluid in the lower portion of a fully-turbulent boundary layer, after an abrupt change in surface roughness.
Abstract: A theory is developed which describes the adjustment of the flow of a hydrostatically neutral fluid in the lower portion of a fully-turbulent boundary layer, after an abrupt change in surface roughness.
The model is based on the hypothesis that the horizontal shear stress is proportional to the turbulent energy. The theory postulates that the flow is primarily governed by the dominant terms of the horizontal-momentum, continuity, and turbulent-energy equations. The model was solved by numerical techniques on a digital computer.
Unlike previous models there are no a priori assumptions about the distribution of velocity or stress, the behaviour of the nondimensional wind shear, mixing length, or momentum-exchange coefficient in the transition region.
The theory, in contrast to earlier theories, suggests the distribution of turbulent energy, as well as velocity. An inflection point is predicted in the transition velocity-profile. The nondimensional wind shear is found to differ significantly from unity in the transition region. These predictions agree with observation.
TL;DR: In this article, experimental results from constant free-stream velocity boundary layers are reported for a variety of constant and slowly varying injection and suction wall conditions, and a description is given of the flow characteristics of these air experiments.
TL;DR: In this paper, a rate equation is proposed to govern the variation of the effective turbulent viscosity, and the effects of generation, convection, diffusion, and decay are each represented by appropriate terms leaving only two empirical constants to be determined by experiment.
Abstract: A rate equation is proposed to govern the variation of the effective turbulent viscosity. The effects of generation, convection, diffusion, and decay are each represented by appropriate terms leaving only two empirical constants to be determined by experiment. This rate equation together with the equations of motion form a closed system applicable to quasiparallel turbulent shear flows. For an incompressible turbulent boundary layer with zero pressure gradient, solutions were obtained by assuming local similarity and a linear growth of the boundary‐layer thickness. Another problem, the turbulent‐nonturbulent interface at the outer edge of the boundary layer was treated by using the further assumption that the large scale motion of the interface has no significant contribution to the Reynolds stress. It can be shown that for a nearly homogeneous domain, Prandtl's mixing length theory is a limiting case of the present theory.
TL;DR: In this article, Schlieren observations of the growth and structural changes in outwardly propagating spherical flames in laminar and isotropic turbulent flow fields are presented and discussed, both in light of the new definition and the cellular instability behaviour of the flame.
TL;DR: In this article, the authors distinguish rigorously between wave motion and turbulence in a stratified fluid, although useful approximations seem feasible, although the task is made more difficult by the fact that properties of turbulence are very little like those described in most theories of turbulence, even when the Reynolds number is quite high.
Abstract: To distinguish rigorously between wave motion and turbulence in a stratified fluid seems impossible, although useful approximations seem feasible. The task is made more difficult by the fact that properties of turbulence are very little like those described in most theories of turbulence, even when the Reynolds number is quite high.
TL;DR: In this article, the Townsend-Bakewell model of the eddies in the wall regions of turbulent shear flows is used to show that viscoelastic fluid properties must lead to significant reductions in the rate of production of turbulent energy.
Abstract: An analysis based on the Townsend-Bakewell model of the eddies in the wall regions of turbulent shear flows shows that viscoelastic fluid properties must lead to significant reductions in the rate of production of turbulent energy. This analysis in turn leads to the proper form of the similarity laws for drag reducing fluids, heretofore deduced empirically.
Measurements of the axial and radial turbulence intensities for flow through smooth round tubes are reported, as are measurements of the time-averaged velocity profiles and the drag coefficients. These indicate that for solutions exhibiting drag reduction at all Reynolds numbers the flow may be transitional to Reynolds numbers of the order of 105. This transitional flow consists of alternating patches of laminar and turbulent fluid, within each of which the flow characteristics are approximately similar to those of Newtonian fluids. At high Reynolds number conditions with the turbulent field fully developed the velocity profile in the core is flatter under drag-reducing conditions than for turbulent Newtonian fluids, a change dependent on the increased isotropy of the turbulent field of the drag-reducing fluid. These effects appear to be a result of increases in the time scales of the radial fluctuations caused by the fluid properties.
Design calculations based upon the present results suggest that in large diameter pipelines, or in boundary layers on large objects, drag reduction may not be attainable under conditions of practical interest until fluids having relaxation times an order of magnitude larger than those presently available, but with comparable viscosity levels, are developed or, alternately, until fluids exhibiting Weissenberg numbers which do not change with deformation rate, can be found.
TL;DR: In this paper, the authors measured mean velocity profiles, longitudinal velocity fluctuations and the wall shearing stress (TW) in two-dimensional accelerated incompressible turbulent boundary layers and found that the wall region adjusts itself to laminar conditions earlier than the outer region.
Abstract: Experiments on reverse transition were conducted in two-dimensional accelerated incompressible turbulent boundary layers. Mean velocity profiles, longitudinal velocity fluctuations and the wall-shearing stress (TW) were measured. The mean velocity profiles show that the wall region adjusts itself to laminar conditions earlier than the outer region. During the reverse transition process, increases in the shape parameter (H) are accompanied by a decrease in the skin friction coefficient (Cf). Profiles of turbulent intensity (u’2) exhibit near similarity in the turbulence decay region. The breakdown of the law of the wall is characterized by the parameter
\[
\Delta_p (=
u[dP/dx]/\rho U^{*3}) = - 0.02,
\]
where U* is the friction velocity. Downstream of this region the decay of fluctuations occurred when the momentum thickness Reynolds number (R) decreased roughly below 400.
TL;DR: In this paper, the effects of a transpiration velocity at the surface of a sphere and of a continuously changing sphere diameter are included, and theoretical corrections for transpiring and shrinking spheres are applied to the mass transfer data for ice spheres that are melting in an agitated brine bath.
Abstract: Mass transfer to spheres suspended in an agitated liquid has been studied both experimentally and theoretically. Finite-difference solutions are obtained for mass transfer from a sphere to a fluid flowing past it in steady viscous flow. The effects of a transpiration velocity at the surface of the sphere and of a continuously changing sphere diameter are included. A normalized presentation of these effects is quite insensitive to the bulk flow Peclet number. When these theoretical corrections for transpiring and shrinking spheres are applied to the mass transfer data for ice spheres that are melting in an agitated brine bath, the corrected mass transfer coefficients are brought into agreement with a generalized correlation published elsewhere. This agreement suggests that the theoretical results apply, with reasonable accuracy, to a shrinking and transpiring sphere that is suspended in a turbulent liquid.
TL;DR: In this paper, the relationship between aerodynamic noise disturbances and boundary-layer transition Reynolds numbers in high-speed wind tunnels was defined and extended to include axisymmetric models, showing a significant increase in the Reynolds numbers with increasing tunnel size.
Abstract: : An experimental investigation of laminar boundary-layer transition on a sharp, 10-deg total-angle, insulated cone at zero yaw was conducted in 12- and 40-in. supersonic wind tunnels at free-stream Mach numbers from 3 to 6. This research was directed toward defining the relationship between the aerodynamic noise disturbances and boundary-layer transition Reynolds numbers in high-speed wind tunnels and extended previously published planar results to include axisymmetric models. A significant increase in the Reynolds numbers with increasing tunnel size (similar to the planar results) is shown to exist. Sharp cone transition Reynolds numbers from ten facilities (12- to 54-in.) for free- stream Mach numbers from 3 to 14 and a unit Reynolds number per inch range from 100,000 to 1,200,000 were correlated using aerodynamic-noise-transition parameters. A quantitative correlation of the ratio between cone and planar Reynolds numbers values was developed which demonstrates a strong Mach number dependence and also indicates a variation with tunnel size and unit Reynolds number.
TL;DR: A Lagrangian formalism for treating the stretching of material line and surface elements convected with a turbulent fluid is developed, and consequences of statistical isotropy for the distribution function of the components of a tensor are derived as mentioned in this paper.
Abstract: A Lagrangian formalism for treating the stretching of material line and surface elements convected with a turbulent fluid is developed, and consequences of statistical isotropy for the distribution function of the components of a tensor are derived. Incompressibility and isotropy are then used to prove rigorously that the expectation value of the logarithmic change of the length of a line element (and area of a surface element) is greater than zero. A connection between the distribution function of the velocity shear tensor and that of the symmetric time‐development tensor used in the Lagrangian formalism is shown.