Showing papers on "Reynolds number published in 1972"

An investigation of particle trajectories in two-phase flow systems

Abstract: This paper describes a theoretical investigation into (i) the response of a spherical particle to a one-dimensional fluid flow, (ii) the motion of a spherical particle in a uniform two-dimensional fluid flow about a circular cylinder and (iii) the motion of a particle about a lifting aerofoil section. In all three cases the drag of the particle is allowed to vary with (instantaneous) Reynolds number by using an analytical approximation to the standard experimental drag-Reynolds-number relationship for spherical particles.

Experiments on the flow past spheres at very high Reynolds numbers

Abstract: The present work is concerned with the flow past spheres in the Reynolds number range 5 × 104 [les ] Re [les ] 6 × 106. Results are reported for the case of a smooth surface. The total drag, the local static pressure and the local skin friction distribution were measured at a turbulence level of about 0·45%. The present results are compared with other available data as far as possible. Information is obtained from the local flow parameters on the positions of boundary-layer transition from laminar to turbulent flow and of boundary-layer separation. Finally the dependence of friction forces on Reynolds number is pointed out.

Numerical Simulation of Three-Dimensional Homogeneous Isotropic Turbulence

Abstract: This Letter reports numerical simulations of three-dimensional homogeneous isotropic turbulence at wind-tunnel Reynolds numbers. The results of the simulations are compared with the predictions of the direct-interaction turbulence theory.

The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles

Abstract: The effect of rotary Brownian motion on the rheology of a dilute suspension of rigid spheroids in shear flow is considered for various limiting cases of the particle aspect ratio r and dimensionless shear rate γ/D. As a preliminary the probability distribution function is calculated for the orientation of a single, axisymmetric particle in steady shear flow, assuming small particle Reynolds number. The result for the case of weak-shear flows, γ/D [Lt ] 1, has been known for many years. After briefly reviewing this limiting case, we present expressions for the case of strong shear where (r3 + r−3) [Lt ] γ/D, and for an intermediate regime relevant for extreme aspect ratios where 1 [Lt ] γ/D [Lt ] (r3 + r−3). The bulk stress is then calculated for these cases, as well as the case of nearly spherical particles r ∼ 1, which has not hitherto been discussed in detail. Various non-Newtonian features of the suspension rheology are discussed in terms of prior continuum mechanical and experimental results.

The structure and stability of vortex rings

Abstract: A series of observations on experimentally produced vortex rings is described. The flow field, ring velocity and growth rate were observed using dye and hydrogen-bubble techniques. It was found that stable rings are formed and grow in such a way that most of their vorticity is distributed throughout a fluid volume which is larger than and moving with the visible dye core.As the vorticity diffuses out of this moving body of fluid into the outer irrotational fluid, it has two effects. It causes some of the fluid, with newly acquired vorticity, to be entrained into the interior of the bubble, while the rest is left behind and accounts for the appearance of ring vorticity in a wake. It was found that the velocity of translation U of these stable rings varies as t−1, at high Reynolds number, where t is the time measured from the start of the motion at a virtual origin at downstream infinity. A simple theoretical model is presented which explains all of these features of the observed stable flow. Rings of even higher Reynolds number become unstable and shed significantly more vorticity into the wake. Under some circumstances a new more stable vortex emerges from this shedding process and continues with less vorticity than before. Eventually, the ring motion ceases as all of its vorticity is deposited into the wake and is spread by viscous diffusion. Observations of the interaction between two nearly identical rings travelling a common path showed that, contrary to popular belief, rings do not pass back and forth through one another, but that the rearward one becomes entrained into the forward one. Only when the rearward ring has a much higher velocity than its partner can it emerge from the joining process and leave a slower-moving ring behind.

Similarity criteria for the application of fluid models to the study of air pollution meteorology

Abstract: Similarity criteria for modeling atmospheric flows in air and water are reviewed. It is shown that five nondimensional parameters plus a set of nondimensional boundary conditions must be matched in model and prototype. The neglect of the Rossby number can lead to serious errors in modeling of diffusion in a prototype with a length scale greater than about five kilometers. The Reynolds number, the Peclet number and the Reynolds-Schmidt product criteria may be neglected if the model flow is of sufficiently high Reynolds number. The Froude number criterion appears to be the most important. The complete specification of boundary conditions is found to be nebulous, but is discussed in some detail. Over-roughening of the model surface may be necessary to satisfy a roughness Reynolds number criterion. Both air and water appear to be suitable fluids to use as modeling media.

Fully developed asymmetric flow in a plane channel

Abstract: The paper presents the results of a detailed experimental examination of fully developed asymmetric flow between parallel planes. The asymmetry was introduced by roughening one of the planes while the other was left smooth; the ratio of the shear stresses at the two surfaces was typically about 4:1.The main emphasis of the research has been on establishing the turbulence structure, particularly in the central region of the channel where the two dissimilar wall flows (generated by the smooth and rough surfaces) interact. Measurements have included profiles of all non-zero double and triple velocity correlations; spectra of the same correlations at several positions in the channel; skewness and flatness factors; and lateral two-point space correlations of the streamwise velocity fluctuation.The region of greatest interaction is characterized by strong diffusional transport of turbulent shear stress and kinetic energy from the rough towards the smooth wall region, giving rise, inter alia, to an appreciable separation between the planes of zero shear stress and maximum mean velocity. The profiles of length scales of the larger-scale motion are, in contrast to the turbulent velocity field, nearly symmetric. Moreover, it appears that at high Reynolds numbers the small-scale motion may in many respects be treated as isotropic.

Mechanism by Which a Two‐Dimensional Roughness Element Induces Boundary‐Layer Transition

Abstract: An experimental investigation of the effect of two‐dimensional roughness elements on boundary‐layer transition is described. Primary emphasis is given to the nature of disturbances within the recovery zone, i.e., that region in the immediate downstream of the roughness where the mean flow has been distorted by the presence of the roughness. Detailed measurements of mean velocity distributions, of disturbance spectra, and intensity, growth, and decay of disturbances at discrete frequencies were made for a range of unit Reynolds numbers. The measurements demonstrate that the behavior can best be understood by considering wave‐type disturbances, and that the basic mechanism by which a two‐dimensional roughness element induces earlier transition to turbulent flow is by the destabilizing influence of the flow within the recovery zone. Comparison with the behavior expected from stability theory supports this conclusion.

Mechanism by which a two-dimensional roughness element induces boundary-layer transition: roughness induced transition

Abstract: An experimental investigation of the effect of two-dimensional roughness elements on boundary layer transition is described. Primary emphasis is given to the nature of disturbances within the recovery zone, i.e., that region in the immediate downstream of the roughness where the mean flow has been distorted by the presence of the roughness. Detailed measurements of mean velocity distributions, of disturbance spectra, and intensity, growth, and decay of disturbances at discrete frequencies were made for a range of unit Reynolds number. The measurements demonstrate that the behavior can best be understood by considering wave-type disturbances, and that the basic mechanism by which a two-dimensional roughness element induces earlier transition to turbulent flow is by the destabilizing influence of the flow within the recovery zone. Comparison with the behavior expected from stability theory supports this conclusion.

An experimental study of the velocity distribution and transition to turbulence in the aorta

Abstract: The development and evaluation of a hot-film probe, suitable for use within arteries and operated with a commercial constant-temperature anemometer and linearizcr, is described. The performance of the system in the recording of arterial velocity wave forms is described, and instantaneous and time-averaged velocity profiles constructed from measurements in the thoracic aorta of dogs are presented. The profiles were blunt, with boundary layers estimated to be less than 2 mm thick throughout the cycle, and significant skews were observed, the explanation for which appears to lie in the influence of local geometry on the flow. A preliminary study of flow disturbances in the aorta based on visual observation of instantaneous velocity wave forms and frequency spectrum analysis is reported. The occurrence of flow disturbances and turbulence is shown to be related to peak Reynolds number and the frequency parameter α. The possible roles of free-stream disturbances and boundary-layer transition in generating these disturbances are discussed.

Measurement of local entrainment rate in the initial region of axisymmetric turbulent air jets

Abstract: The local entrainment rate in the initial region of axisymmetric turbulent air jets has been measured by a novel method, which is an adaptation of the ‘porous-wall’ technique used by Ricou & Spalding (1961). The local entrainment rate, which is independent of the nozzle Reynolds number for values greater than 6 × 104, is strongly dependent upon the axial distance. At an axial distance of one nozzle diameter the local entrainment rate is only about one-third of that in the fully developed jet; the entrainment rate increases with increasing axial distance to reach the fully developed value at an axial distance of about thirteen nozzle diameters.

The Hemodynamic Importance of the Geometry of Bifurcations in the Circle of Willis (Glass Model Studies)

Abstract: The critical Reynolds number, Rec, at which turbulence developed in glass model bifurcations was measured with an Evans blue indicator for bifurcations with a branch/trunk area ratio of unity, and bifurcation angles of 45°, 90°, 135°, and 180°. The Rec dropped from 2,500 in a straight tube to 1,200 in the 180° bifurcation. Further drops occured with pulsatile flow (if the mean flow rate was used to calculate the velocity). Three sizes of aneurysms at the apex of the 90° bifurcation lowered the Rec at small bifurcations, and less in the 180° ones. The curves for steady and pulsatile flow crossed at 135°. We did qualitative, but not quantitative, assessments of axial stream impingement on the apex of the bifurcation in the site of aneurysm formation, and of boundary layer separation and vortex shedding at the lateral angles. Both appeared to vary with the angle of the bifurcation and the Reynolds number. We also studied flow profiles in glass models of anterior cerebral-anterior communicating artery bifurca...

Cavity flows driven by buoyancy and shear

Abstract: Fluid motion driven by the combined effects of a moving wall and natura convection is examined for rectangular cavities with heightlwidth ratios of ½, 1 and 2. The Reynolds number and Prandtl number are held fixed at Re = 100 and Pr = 1; the Grashof number is varied over the range of values Gr = 0, ±104, ±106. Flow and temperature fields obtained from numerical solutions of the Navier-Stokes equations reveal a marked influence of buoyancy for the larger aspect ratios when Gr = ±106 and the dominance of buoyancy for all aspect ratios when Gr = ± 106. Results are compared with earlier work where possible and some observations are offered on the convergence of the numerical solutions.

Shallow Water Waves on a Viscous Fluid—The Undular Bore

Abstract: A theory is presented for the surface profile above a fully developed Poiseuille channel flow. Small disturbances to this flow are examined, and it is shown that if the (channel depth)/(wavelength) ratio is small (shallow waves), and the Reynolds number large enough, these disturbances initially travel at the classical dynamic (Burns) wave speeds. However, by introducing appropriate far‐field coordinates it follows that the disturbance eventually travels at a different wave speed—the kinematic wave speed. To confirm this, the dynamic waves are shown to decay by using standard boundary layer techniques. This general result (of decay) agrees with previous one‐dimensional theories. The profile close to the kinematic wave front is examined and shown to satisfy an equation of the form ηT + ηηX + ηXXX = ΔηXX, where η(X, T) is the surface profile. This equation is called the Korteweg‐de Vries‐Burgers equation. The form of the steady solution of this equation exhibits all the characteristics of the undular bore. A bound on Δ agrees with stability requirements found by other authors using different methods.

Numerical simulation of turbulence

Abstract: Numerical simulations of three-dimensional homogeneous, isotropic turbulence at windtunnel Reynolds numbers (Rλ 20–40) are reported. The results of the simulations are compared with the predictions of turbulence theories.

Some properties of sink-flow turbulent boundary layers

Abstract: An experimental study of asymptotic sink-flow turbulent boundary layers is reported. Three levels of acceleration corresponding to values of the acceleration parameter K of 1·5 × 10−6, 2·5 × 10×6 and 3·0 × 10×6 have been examined. In addition to mean velocity profiles, measurements have been obtained of the profiles of longitudinal turbulence intensity, and, for the lowest value of K, of the lateral and transverse components as well. Measurements at selected positions in the boundary layer of the power spectral density indicate that none of the boundary layers exhibit an inertial subrange; for the steepest acceleration, in particular, throughout the boundary layer the spectrum shapes are similar in form to those reported within the viscous sublayer of a high Reynolds number turbulent flow.

A nonlinear instability burst in plane parallel flow

Abstract: An infinitesimal centre disturbance is imposed on a fully Ldveloped plane Poiseuille flow at a Reynolds number R slightly greater than the critical value Rc for instability. After a long time, t, the disturbance consists of a modulated wave whose amplitude A is a slowly varying function of position and time. In an earlier paper (Stewartson & Stuart 1971) the parabolic differential equation satisfied by A for two-dimensional disturbances was found; the theory is here extended to three dimensions. Although the coefficients of the equation are coinples, a start is made on elucidating the properties of its solutions by assuming that these coefficients are real. It is then found numerically and confirmed analytically that, for a finite value of (R-Rc)t, the amplitude A develops an infinite peak at the wave centre. The possible relevance of this work to the phenomenon of transition is discussed.

A note on von Kármán's constant in low Reynolds number turbulent flows

Abstract: An analysis of existing data on low Reynolds number flows strongly suggests that the conclusion of Simpson (1970) concerning the variation of von Karmas constant κ with Reynolds number is not correct. This implies that Coles’ (1962) assumption of the validity of the logarithmic velocity profile at low Reynolds numbers is correct and, moreover, that the inference drawn by Coles and later authors regarding the presence of viscous effects in the outer layer is valid. The analysis shows that these viscous effects are not present in duct flows, so that they are presumably associated with the presence of a turbulent-irrotational interface; it is argued that the ‘viscous superlayer’ can affect a large part of the outer layer at low Reynolds numbers. The data analysis incidentally shows that the viscous sublayer is more strongly affected by shear-stress gradients or transverse wall curvature than is the rest of the inner layer.

Cavity and Wake Flows

Abstract: The phenomenon of wake formation behind a body moving through a fluid, and the associated resistance of fluids, must have been one of the oldest experiences of man. From an analytical point of view, it is also one of the most difficult problems in fluid mechanics. Rayleigh, in his 1876 paper, observed that "there is no part of hydrodynamics more perplexing to the student than that which treats of the resistance of fluids." This insight of Rayleigh is so penetrating that the march of time has virtually left no mark on its validity even today, and likely still for some time to come. The first major step concerning the resistance of fluids was made over a century ago when Kirchhoff (1869) introduced an idealized inviscid-flow model with free streamlines (or surfaces of discontinuity) and employed (for steady, plane flows) the ingenious conformal-mapping technique that had been invented a short time earlier by Helmholtz (1868) for treating two-dimensional jets formed by free streamlines. This pioneering work offered an alternative to the classical paradox of D’Alembert (or the absence of resistance) and laid the foundation of the free-streamline theory. We appreciate the profound insight of these celebrated works even more when we consider that their basic idea about wakes and jets, based on a construction with surfaces of discontinuity, was formed decades before laminar and turbulent flows were distinguished by Reynolds (1883), and long before the fundamental concepts of boundary-layer theory and flow separation were established by Prandtl (1904a). However, there have been some questions raised in the past, and still today, about the validity of the Kirchhoff flow for the approximate calculation of resistance. Historically there is little doubt that in constructing the flow model Kirchhoff was thinking of the wake in a single-phase fluid, and not at all of the vapor-gas cavity in a liquid; hence the arguments, both for and against the Kirchhoff flow, should be viewed in this light. On this basis, an important observation was made by Sir William Thomson, later Lord Kelvin (see Rayleigh 1876) "that motions involving a surface of separation are unstable" (we infer that instability here includes the viscous effect). Regarding this comment Rayleigh asked "whether the calculations of resistance are materially affected by this circumstance as the pressures experienced must be nearly independent of what happens at some distance in the rear of the obstacle, where the instability would first begin to manifest itself." This discussion undoubtedly widened the original scope, brought the wake analysis closer to reality, and hence should influence the course of further developments. An expanded discussion essentially along these lines was given by Levi-Civita (1907) and was included in the survey by Goldstein (1969). Another point of fundamental importance is whether the Kirchhoff flow is the only correct Euler (or outer) limit of the Navier-Stokes solution to steady flow at high Reynolds numbers. If so, then a second difficulty arises, a consequence of the following argument: We know that the width of the Kirchhoff wake grows parabolically with the downstream distance x, at a rate independent of the (kinematic) viscosity u. If Prandtl’s boundary-layer theory is then applied to smooth out the discontinuity (i.e. the vortex sheet) between the wake and the potential flow, one obtains a laminar shear layer whose thickness grows like (ux/U)^-1/2 in a free stream of velocity U. Hence, for sufficiently small u/U the shear layers do not meet, so that the wake bubble remains infinitely long at a finite Reynolds number, a result not supported by experience. (For more details see Lagerstrom 1964, before p. 106, 131; Kaplun 1967, Part II.) The weaknesses in the above argument appear to lie in the two primary suppositions that, first, the free shear layer enveloping the wake would remain stable indefinitely, and second (perhaps a less serious one), the boundary-layer approximation would be valid along the infinitely long wake boundary. Reattachment of two turbulent shear layers, for instance, is possible since their thickness grows linearly with x. By and large, various criticisms, of the Kirchhoff flow model have led to constructive refinements of the free-streamline theory rather than to a weakening of the foundation of the theory as a valuable idealization. The major development in this direction has been based on the observation that the wake bubble is finite in size at high Reynolds numbers. (The wake bubble, or the near-wake, means, in the ordinary physical sense, the region of closed streamlines behind the body as characterized by a constant or nearly constant pressure.) To facilitate the mathematical analysis of flows with a finite wake bubble, a number of potential-flow models have been introduced to give the near-wake a definite configuration as an approximation to the inviscid outer flow. These theoretical models will be discussed explicitly later. It suffices to note here that all these models, even though artificial to various degrees, are aimed at admitting the near-wake pressure coefficient as a single free parameter of the flow, thus providing a satisfactory solution to the state of motion in the near part of the wake attached to the body. On the whole, their utility is established by their capability of bringing the results of potential theory of inviscid flows into better agreement with experimental measurements in fluids of small viscosity. The cavity flow also has a long, active history. Already in 1754, Euler, in connection with his study of turbines, realized that vapor cavitation may likely occur in a water stream at high speeds. In investigating the cause of the racing of a ship propeller, Reynolds (1873) observed the phenomenon of cavitation at the propeller blades. After the turn of this century, numerous investigations of cavitation and cavity flows were stimulated by studies of ship propellers, turbomachinery, hydrofoils, and other engineering developments. Important concepts in this subject began to appear about fifty years ago. In an extensive study of the cavitation of water turbines, Thoma (1926) introduced the cavitation number (the underpressure coefficient of the vapor phase) as the principal similarity parameter, which has ever since played a central role in small-bubble cavitation as well as in well-developed cavity flows. Applications of free-streamline theory to finite-cavity flows have attracted much mathematical interest and also provided valuable information for engineering purposes. Although the wake interpretation of the flow models used to be standard, experimental verifications generally indicate that the theoretical predictions by these finite-wake models are satisfactory to the same degree for both wake and cavity flows. This fact, however, has not been widely recognized and some confusion still exists. As a possible explanation, it is quite plausible that even for the wake in a single-phase flow, the kinetic energy of the viscous flow within the wake bubble is small, thus keeping the pressure almost unchanged throughout. Although this review gives more emphasis to cavity flows, several basic aspects of cavity and wake flows can be effectively discussed together since they are found to have many important features in common, or in close analogy. This is in spite of relatively minor differences that arise from new physical effects, such as gravity, surface tension, thermodynamics of phase transition, density ratio and viscosity ratio of the two phases, etc., that are intrinsic only to cavity flows. Based on this approach, attempts will be made to give a brief survey of the physical background, a general discussion of the free-streamline theory, some comments on the problems and issues of current interest, and to point out some basic problems yet to be resolved. In view of the vast scope of this subject and the voluminous literature, efforts will not be aimed at completeness, but rather on selective interests. Extensive review of the literature up to the 1960s may be found in recent expositions by Birkhoff & Zarantonello (1957), Gilbarg (1960), Gurevich (1961), Wehausen (1965), Sedov (1966), Wu (1968), Robertson & Wislicenus (1969), and (1961).

Some effects of intense turbulence on the aerodynamics of a circular cylinder at subcritical Reynolds number

Abstract: The effect, of high intensity large-scale free-stream turbulence on the flow past a rigid circular cylinder has been studied experimentally a t subcritical Reynolds numbers. Grids were used to produce homogeneous turbulence fields with longitudinal scales ranging from 0·36 to 4·40 cylinder diameters and with longitudinal intensities greater than 10%. Power and cross-spectra of the turbulence components (the ‘system input’) have been measured in order to carefully define the turbulence characteristics.In the response experiments, a special model measured arbitrary two-point pressure correlations. Subsequent integrations yielded the specbral properties of the unsteady lift and drag. Measurements of mean drag and Strouhal frequency indicate that to some extent even severe large-scale turbulence can be considered to be qualitatively equivalent to an increase in the effective Reynolds number. Vortex shedding is not seriously disrupted by severe turbulence, but is affected more by low than by high frequencies. The unsteady lift response is still dominated by the vortex shedding, whereas the unsteady drag becomes primarily a response to turbulence. The cross-spectra of the drag forces for the one turbulence case examined overlay well when plotted against lateral separation divided by wavelength. This has enabled a ‘describing function’ for the drag response to turbulence to be derived. This describing function is the central element needed for the calculation of the structural response of such cylinders in the drag direction.

Numerical Solution of the Navier‐Stokes Equations by the Finite Element Method

Abstract: Occurrences of viscous fluid flows in arbitrary internal passages are numerous, an analytical solution of the governing Navier‐Stokes equations cannot be obtained. Even a numerical approach faces difficulties arising from the nonlinearity and complexity of the geometry involved. To remedy these difficulties, the time‐dependent Navier‐Stokes equations are solved by the finite element method wherein a steady‐state solution is assumed when the time‐dependent solution becomes convergent. A family of locally constricted channels was considered in the computations, and in each case, the shear stress at the wall was found to be sharply increased at and near the region of constriction. Computations were carried out to Reynolds numbers when a separation eddy was established. The numerical scheme used seems to be fairly stable.

The stability of Poiseuille flow in a pipe of circular cross-section

Abstract: The stability of Poiseuille flow in a pipe of circular cross-section to azimuthally varying as well as axisymmetric disturbances has been studied. The perturbation velocity and pressure were expanded in a complete set of orthonormal functions which satisfy the boundary conditions. Truncating the expansion yielded a matrix differential equation for the time dependence of the expansion coefficients. The stability characteristics were determined from the eigenvalues of the matrix, which were calculated numerically. Calculations were carried out for the azimuthal wavenumbers n = 0,…, 5, axial wavenumbers α between 0·1 and 10·0 and αR [les ] 50000, R being the Reynolds number. Our results show that pipe flow is stable to infinitesimal disturbances for all values of α, R and n in these ranges.

Linear spatial stability of pipe Poiseuille flow

Abstract: A theoretical study of the spatial stability of Poiseuille flow in a rigid pipe to infinitesimal disturbances is presented. Both axisymmetric and non-axisymmetric disturbances are considered. The coupled, linear, ordinary differential equations governing the propagation of a disturbance that has a constant frequency and is imposed at a specified location in the fluid are solved numerically for the complex eigenvalues, or wavenumbers, each of which defines a mode of propagation. A series solution for small values of the pipe radius is derived and step-by-step integration to the pipe wall is then performed. In order to ascertain the number of eigenvalues within a closed region, an eigenvalue search technique is used. Results are obtained for Reynolds numbers up to 10000. For these Reynolds numbers it is found that the pipe Poiseuille flow is spatially stable to all infinitesimal disturbances.

Laminar dispersion in curved tubes and channels

Abstract: Dispersion in curved tubes and channels is treated analytically, using the velocity distribution of Topakoglu (1967) for tubes and that of Goldstein (1965) for curved channels. The result for curved tubes is compared with that obtained previously by Erdogan & Chatwin (1967) and it is found that the presentdispersion coefficient contains the Erdogan & Chatwin result as a limiting case.The most striking difference between the results is that Erdogan & Chatwin predict that the dispersion coefficient is always decreased by curvature if the Schmidt number exceeds 0.124, which is the ease for essentially all systems of practical interest. In contrast, the present result, equation (76), predicts that the dispersion coefficient may be increased substantially by curvature in low Reynolds number flows, particularly in liquid systems which would be of interest in biological systems.Two competing mechanisms of dispersion are present in curved systems. Curvature increases the variation in residence time across the flow in comparison with straight systems and this in turn increases the dispersion coefficient. The secondary flow which occurs in curved tubes creates a transverse mixing which decreases the dispersion coefficient. The results demonstrate that the relative importance of these two effects changes with the Reynolds number, since the dispersion coefficient first increases and then decreases as the Reynolds number increases. Since secondary flows are not present in curved channels the dispersion coefficient is increased over that in straight channels for all cases.