Showing papers on "Boundary layer published in 1977"

Momentum transfer in boundary layers

Abstract: Too many books on fluid mechanics are written on the strength of original material which makes up one chapter or less; the remainder being a restatement of the familiar contents of other books. This book by Cebeci and Bradshaw is totally free from such objections. The authors have both made very substantial contributions to knowledge in the area covered by the book. This shows clearly in the authority of the presentation, in the large number of very recent references and in the material, which is fresh and up-to-date without in any way being self-consciously trendy. Despite its title, the book is concerned with thin shear layers of all kinds and for the most part does not discuss them explicitly in terms of momentum transport: the title is presumably chosen to show that heat and mass transfer effects are not considered. The material covered is an impressive array of cases, representing the present state of knowledge. Far from the conventional concentration on two-dimensional wall boundary layers, the book gives considerable emphasis to both external and internal flows, to axisymmetric and three dimensional configurations and to free shear layers. Modern finite difference methods are described and, where appropriate, listings of Fortran programs are presented. The emphasis is on practical cases and practical methods of solution. A large number of novel and interesting problems and worked examples are given. The reviewer found the exposition uneven: in many parts, every comment carries an exciting new insight but in others the vision seems diminished and the presentation difficult to follow. A few passages consist of the presentation of the first author's research with little or no discussion of other methods. Surprisingly the derivation of the equations makes no mention of matched asymptotic expansions. Indeed, the laminar thin shear layer equations are accorded only the somewhat humble status of a data correlation. However, it would be unfair and misleading to emphasise minor shortcomings. The book is a powerful addition to the literature. It wil l demand an effort from its readers and is likely to be too advanced and uncompromising for undergraduate use but for postgraduate students and research workers, it is very welcome indeed. Any effort it demands is amply repaid. B. L. HUNT

A Correlating Equation for Forced Convection From Gases and Liquids to a Circular Cylinder in Crossflow

Abstract: A single comprehensive equation is developed for the rate of heat and mass transfer from a circular cylinder in crossflow, covering a complete range of Pr (or Sc) and the entire range of Re for which data are available. This expression is a lower bound (except possibly for RePr < 0.2); free-stream turbulence, end effects, channel blockage, free convection, etc., may increase the rate. In the complete absence of free convection, the theoretical expression of Nakai and Okazaki may be more accurate for RePr < 0.2. The correlating equation is based on theoretical results for the effect of Pr in the laminar boundary layer, and on both theoretical and experimental results for the effect of Re. The process of correlation reveals the need for theoretical results for the effect of Pr in the region of the wake. Additional experimental data for the effect of Pr at small Pe and for the effect of Re during the transition in the point of separation are also needed.

Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike

Abstract: An analysis is made for steady free convection about a vertical flat plate embedded in a saturated porous medium at high Rayleigh numbers. Within the framework of boundary layer approximations, similarity solutions are obtained for a class of problems where wall temperature varies as xλ, i.e., a power function of distance from the origin where wall temperature begins to deviate from that of the surrounding fluids. Analytical expressions are obtained for boundary layer thickness, local and overall surface heat flux, and local and average heat transfer coefficients. Application to convective heat transfer about an isothermal dike intruded in an aquifer is discussed.

A Higher Order Closure Model for Canopy Flow

Abstract: The equations of motion were used to develop a one-dimensional, nonbuoyant mathematical model of air flow within vegetative canopies. The model consists of equations for mean horizontal momentum, Reynolds stress, and for the three components of turbulent kinetic energy with closure achieved by parameterizing the higher order terms. This eliminates the need to model the Reynolds stress directly using an eddy viscosity. The closure schemes rely upon a prescribed length scale and have been used elsewhere in modeling the atmospheric boundary layer free of vegetation. The equations were solved numerically using specified boundary conditions. Using a profile of plant area density for a crop of corn (Zea mays L.) the model predicted mean wind velocity, Reynolds stress and turbulent intensities for the region from the soil surface to twice the canopy height that compare well with experimental measurements (Shaw et al., 1974a,b). The model is believed to overestimate the intensity of turbulence generated ...

Large structure in a turbulent boundary layer

Abstract: Using an array of hot wires and wall shear stress probes results are presented which support the hypothesis that an organized structure exists in the boundary layer. From the correlation between wall shear stress and velocity it is shown that this structure is at an oblique angle to the free stream. The inferred angle from this correlation is found to agree remarkably with recent flow visualization photographs. Evidence is found that the large scale motion in the organized structure produces a slowly varying component in the wall shear stress and, importantly, a high frequency large amplitude fluctuation occurring near the maximum in the slowly varying wall shear. It is suggested that this latter fluctuation is associated with the bursting phenomenon. A model which accounts for these results is described. It leads to a description of the bursting process in terms of a rotational instability and while too approximate to give more than good order of magnitude estimates, it gives values of the wall streak spacing which are not inconsistent with previous measurements.

Subgrid-Scale Condensation in Models of Nonprecipitating Clouds

Abstract: One of the shortcomings of present condensation schemes is the assumption that a computational grid volume is either entirely saturated or entirely unsaturated, which is a crude approximation in some instances even with a relatively fine mesh. The concept of a statistical distribution of the points where condensation occurs inside a given grid volume is discussed and results are presented for diagnosing both the fraction of the grid volume containing saturated air and the liquid water content when that fraction is less than unity. An abbreviated procedure is tested in a boundary layer model of 50 m mesh for a case of nonprecipitating tropical moist convection. Two main effects upon the simulated turbulent field are found: first, the cloud activity measured by the total liquid water content or by the vertical moisture transport is increased and extends farther up, at the expense of subcloud layer moisture; second, the temporal variations within the model domain of the calculated turbulent properti...

Three-Dimensional Flow Within a Turbine Cascade Passage

Abstract: Detailed measurements of the subsonic flow in a large-scale, plane turbine cascade were made to evaluate the three-dimensional nature of the flow field. Tests were conducted at a passage aspect ratio of 1.0 with a collateral inlet boundary layer. Flow visualization was done on airfoil and endwall surfaces. Velocity and pressure measurements were taken before and behind the cascade and in six axial planes within a cascade passage, using a five-hole probe. Hot wire measurements were taken in the endwall boundary layer within the cascade passage. The characteristics of the endwall boundary layer are presented, showing that three-dimensional separation is an important feature of end-wall flow. A large part of the endwall boundary layer was found to be very thin when compared to the cascade inlet boundary layer. Data showing the growth of aerodynamic loss through the passage are discussed.

Compressible Turbulent Shear Layers

Abstract: It is generally accepted that the direct effects of density fluctuations on turbulence are small if the root-mean-square density fluctuation is small compared with the absolute density: this is Morkovin's hypothesis (Favre 1964, p. 367). This means that the turbulence structure of boundary layers and wakes at free-stream Mach numbers Me less than about 5, and of jets at Mach numbers less than about 1.5, is closely the same as in the corrcsponding constant-density flow. By "turbulence structure" we mean dimensionless properties like correlation coefficients and spectrum shapes: the skin-friction coefficient cf == Tw/tPeU; and other ratios of turbulence quantities to mean flow quantities are greatly affected by the influence of mean density changes on the mean motion. The effect of mean density variations in x or y on the turbulence structure is not covered by Morkovin's hypothesis, but is often negligible at the lower Mach numbers if stream wise pressure gradients are small. Therefore assumptions about turbulence structure that give good results in calculation methods for constant-density /low will, if properly scaled, give good results in compressible boundary layers or wakes for Me 5, say. Basic equations for compressible shear layers are given by Howarth (1953) and Lin (1959). More recent treatments (FavJe 1971, Cebeci & Smith 1974, Rubesin & Rose 1973, Bilger 1975) use "mass-weighted" variables, which remove density iluctuations from the time-averaged equations of motion but not from the turbulence or from the response of measuring instruments (although Laufer, in Birch et aI1972, p. 462, suggests that pitot tubes probably yield mass-averaged velocities). It seems probable that the difference between conventional and mass-weighted averages rises more slowly with Mach number than current errors in measuring either. Of problems 1 The author is grateful for a number of helpful comments or contributions, especialIy

Coherent motions in the outer region of turbulent boundary layers

Abstract: The technique of simultaneous hot‐wire anemometry and flow visualization has been used to determine the average characteristics of two important scales of motion in the outer region of turbulent boundary layers: large scale motions (average length 1.6δ), and ’’typical eddy’’ motions (average streamwise length approximately 200 ν/u τ). Results showed that the Reynolds number dependent ’’typical eddies’’ produced most of the Reynolds stress in the outer half of the layer at R ϑ≈1200, and that they are formed on the upstream side of large scale motions at all Reynolds numbers investigated. This phase relationship explains the scaling of the frequency of occurrence of outer layer bursts (which are identified with ’’typical eddies’’) on the free stream velocity and overall boundary layer thickness, although it is found that the lengths of the ’’typical eddies’’ scale on inner layer variables. In the log region, roughly one‐half of the large scale motions sampled had zone‐averaged streamwise velocity defects. Ensemble averaged results showed that they were associated with significant Reynolds stress contributions. A structural model showing the phase relationship of ’’typical eddies’’ and large scale motions is presented.

Analysis of the development of dynamic stall based on oscillating airfoil experiments

Abstract: The effects of dynamic stall on airfoils oscillating in pitch were investigated by experimentally determining the viscous and inviscid characteristics of the airflow on the NACA 0012 airfoil and on several leading-edge modifications. The test parameters included a wide range of frequencies, Reynolds numbers, and amplitudes-of-oscillation. Three distinct types of separation development were observed within the boundary layer, each leading to classical dynamic stall. The NACA 0012 airfoil is shown to stall by the mechanism of abrupt turbulent leading-edge separation. A detailed step-by-step analysis of the events leading to dynamic stall, and of the results of the stall process, is presented for each of these three types of stall. Techniques for flow analysis in the dynamic stall environment are discussed. A method is presented that reduces most of the oscillating airfoil normal force and pitching-moment data to a single curve, independent of frequency or Reynolds number.

Gas transfer to and across an air-water interface

Abstract: A treatment based on Reichardt's formulation of the velocity profile in turbulent flow over a smooth plane surface is shown to give good agreement with published data on the boundary layer transfer of heat and mass over a wide range of Prandtl (or Schmidt) number, s. Applied to transfer to a water surface, agreement with published laboratory results is also good for low air speeds (smooth water). Comparison with observations for the sea shows there is little difference between the calculated evaporation coefficient and those reported for the sea with winds of ∼7 m s -1 . This is consistent with sea trials having so far detected little increase of evaporation and heat transfer coefficients with wind speed over the range 4–10 m s -1 . The air/water transfer of non-reactive gas is governed by the resistance of the viscous sublayer of water, and the smooth surface treatment gives the transfer velocity ( V L ) on a liquid phase basis as V L = 0.082 ( p a /p w ) 1/2 s -2/3 u* where p a p w is the density ratio, air/water, and u * = friction velocity of the surface air flow. The agreement between this formula and published wind tunnel results is good for the smooth water condition. At higher wind speeds transfer exceeds the calculated value and appears then to increase roughly in proportion to the square of the wind speed. DOI: 10.1111/j.2153-3490.1977.tb00746.x

A Realistic Model of the Wind-Induced Ekman Boundary Layer

Abstract: A physically realistic and general model for the vertical eddy viscosity in a homogeneous fluid is proposed. For an infinitely deep ocean the vertical eddy viscosity increases linearly with depth from a value of zero at the free surface. Based on this model a general theory is developed for the drift current resulting from a time-varying surface shear stress. Explicit expressions are given for the temporal development of the drift current in the vicinity of the free surface and for the steady-state response to a suddenly applied uniform shear stress. The steady-state solution predicts the effective Ekman layer depth to be proportional to the square root of the wind shear stress and reproduces the experimentally observed logarithmic velocity deficit near the free surface. The angle between the surface drift current and the wind stress is found to be somewhat smaller (of the order 10°) than predicted by Ekman's classical solution. For the unsteady response to a suddenly applied wind stress the pres...

The calculation of turbulent boundary layers on spinning and curved surfaces

Abstract: The development of turbulent shear layers on rotating or curved surfaces is usually characterized by strong effects of streamline curvature on the turbulence structure. The present contribution deals with the calculation of these effects with a model of turbulence which solves transport equations for the turbulence kinetic energy and its local rate of dissipation. The direct effect of curvature in the model is limited to a single empirical coefficient whose magnitude is directly proportional to a Richardson number based on a time scale of the energy-containing eddies. (In the absence of significant streamline curvature the model reduces to a form that has earlier been extensively tested in various thin shear flows.) Finite difference computations are reported of the following turbulent flows: the boundary layer on concave and convex surfaces; fully developed flow in a curved channel; axisymmetric flow over a spinning cylinder; and heat and mass transfer due to spinning cones of various vertex angles. Agreement with experiment is satisfactorily close in all these cases.

Unsteady flows in a semi-infinite contracting or expanding pipe

Abstract: Physiological pumps produce flows by alternate contraction and expansion of the vessel. When muscles start to squeeze its wall the valve at the upstream end is closed and that at the downstream end is opened, and the fluid is pumped out in the downstream direction. These systems can be modelled by a semi-infinite pipe with one end closed by a compliant membrane which prevents only axial motion of the fluid, leaving radial motion completely unrestricted. In the present paper an exact similar solution of the Navier–Stokes equation for unsteady flow is a semi-infinite contracting or expanding circular pipe is calculated and reveals the following characteristics of this type of flow. In a contracting pipe the effects of viscosity are limited to a thin boundary layer attached to the wall, which becomes thinner for higher Reynolds numbers. In an expanding pipe the flow adjacent to the wall is highly retarded and eventually reverses at Reynolds numbers above a critical value. The pressure gradient along the axis of pipe is favourable for a contracting wall, while it is adverse for an expanding wall in most cases. These solutions are valid down to the state of a completely collapsed pipe, since the nonlinearity is retained in full. The results of the present theory may be applied to the unsteady flow produced by a certain class of forced contractions and expansions of a valved vein or a thin bronchial tube.

Upstream influence on the near field of a plane turbulent jet

Abstract: The effects of the mean and turbulence characteristics of the upstream (initial) boundary layer on the evolution of the flow in the near field of a plane jet have been experimentally investigated for four initial conditions. Rates of jet widening and centerline mean velocity decay as well as the kinematic and geometric virtual origins show evidence of systematic dependence on initial conditions. The growth rate of longitudinal turbulence intensity, and the mass flux are higher when the initial boundary layer is laminar than when turbulent. Immediately downstream of the exit, the nondimensional entrainment rates for the laminar initial boundary layer cases reach peak values which are about twice the delayed peak values for the fully turbulent initial boundary layer cases. Within the first 40 slit widths, increases in total average streamwise momentum flux range from 20% to 56%, the larger increases occurring for the laminar initial boundary layers; about 10% of each increase is due to the turbulence field. While such increases violate the traditionally accepted momentum flux invariance, they are consistent with the negative mean static pressure data.

The nocturnal jet

Abstract: Three case studies of the nocturnal jet at dawn are presented. Observations were made over southern England where the terrain is considerably less homogeneous than that over which previous published data concerning the jet have been obtained, for example the Great Plains and Wangara. It is shown that previously suggested layer models of the boundary layer can be usefully quantified to give the diurnal evolution of a layer-average wind. This model, which involves a decoupling of the flow in middle levels from surface constraints by thermal stratification at night, exhibits a nocturnal jet. Two surface-stress parameterizations, which make the stress proportional to the velocity and to the velocity squared, give a similar wind evolution. The observed abrupt transition in boundary layer structure soon after sunrise is studied by considering the growth and transfer characteristics of perturbations to a homogeneous Boussinesq fluid, which is stratified and possesses an idealized jet wind structure. Depending on the value of an average Richardson number, Ri, two regimes of dynamically unstable eddies are indicated. When +0.35 ≳ Ri −0.03 the preferred eddies are oriented transverse to the shear and when Ri ≲−0.03 they are oriented parallel to the shear. the latter regime, associated with longitudinal convective circulations, is efficient at smoothing the jet momentum distribution to give the characteristic well-mixed daytime wind profile.

Effect of compliant wall motion on turbulent boundary layers

Abstract: A critical analysis of available compliant wall data which indicated drag reduction under turbulent boundary layers is presented. Detailed structural dynamic calculations suggest that the surfaces responded in a resonant, rather than a compliant, manner. Alternate explanations are given for drag reductions observed in two classes of experiments: (1) flexible pipe flows and (2) water−backed membranes in air. Analysis indicates that the wall motion for the remaining data is typified by short wavelengths in agreement with the requirements of a possible compliant wall drag reduction mechanism recently suggested by Langley.

Heat thermal structure in the interfacial boundary layer measured in an open tank of water in turbulent free convection

Abstract: The thermal structure in the boundary layer and its relation to the heat flux from the cooling and evaporating surface of a deep tank of water are investigated. When a deep layer of water in contact with still air above loses heat to the air, the cooled water in a region just under the surface converges along lines and then plunges down in sheets. These sheets of falling water dissipate as they move into the body of the water, which is in turbulent motion. The vertical profiles of the horizontally averaged temperature and its standard deviation agree fairly closely with theoretical profiles based on time averages of the solution to the heat diffusion equation. The differences between observed and thus predicted profile shapes are consistent with the expected effects of the falling cold thermals and the warm return flow, which are neglected in the theories. The profiles of the standard deviation have large values up to the interface and lie between predictions based on boundary conditions of constant surface temperature and constant heat flux, in keeping with the experimental conditions.The relation between the net heat flux and the temperature difference across the boundary layer is given in non-dimensional form by N = 0[sdot ]156R0[sdot ]33, which is in good agreement with the asymptotic similarity prediction N [vprop ] R1/3 but lower than theoretical calculations of the upper bound of N vs. R.

Temperature dissipation fluctuations in a turbulent boundary layer

Abstract: All three components of the dissipation rate of the fluctuating temperature ϑ are measured simultaneously in the inner region of a fully developed turbulent boundary layer at a moderate Reynolds number. Measurements are made with a probe of four cold wires consisting of two closely spaced parallel vertical wires mounted a small distance upstream of two closely spaced parallel horizontal wires. In the inner region of the layer, local isotropy is not closely approximated (∂ϑ/∂z)2〉 (∂ϑ/∂y)2〉 (∂ϑ/∂x)2. The spectral density of the sum χ[= (∂ϑ/∂x)2+(∂ϑ/∂y)2+ (∂ϑ/∂z)2] is similar in shape to that of (∂ϑ/∂y)2 or (∂ϑ/∂z)2 but not as rich in high frequency content as that of (∂ϑ/∂x)2. The probability density of χ has a lower skewness and flatness factor and is more closely log‐normal than those of the individual components. This is true regardless of whether χ and its components are unaveraged or locally averaged over a linear dimension r. When averaging is applied, departures from log‐normality are diminished but ...

Features of a separating turbulent boundary layer in the vicinity of separation

Abstract: Laser anemometer measurements using a directionally sensitive system were obtained for a nominally two-dimensional separating turbulent boundary layer produced by an adverse pressure gradient. An airfoil-type flow was generated in which the flow was accelerated and then decelerated until separation. The results include the skin friction, mean velocity profiles, turbulent shear stresses and intensities, spectra, dissipation rate, turbulent/non-turbulent interfacial intermittency, and eddy speeds.

Microscale temperature and velocity spectra in the atmospheric boundary layer

Abstract: High-frequency fluctuations in temperature and velocity were measured at a height of 2 m above a harvested, nearly level field of rye grass. Conditions were both stably and unstably stratified. Reynolds numbers ranged from 370000 to 740000. Measurements of velocity were made with a hot-wire anemometer and measurements of temperature with a platinum resistance element which had a diameter of 0[sdot ]5 μm and a length of 1 mm. Thirteen runs ranging in length from 78 to 238 s were analysed.Spectra of velocity fluctuations are consistent with previously reported universal forms. Spectra of temperature, however, exhibit an increase in slope with increasing wavenumber as the maximum in the one-dimensional dissipation spectrum is approached. The peak of the one-dimensional dissipation spectrum for temperature fluctuations occurs at a higher wavenumber than that of simultaneous spectra of the dissipation of velocity fluctuations. It is suggested that the change in slope of the temperature spectra and the dissimilarity between temperature and velocity spectra may be due to spatial dissimilarity in the dissipation of temperature and velocity fluctuations. The temperature spectra are compared with a theoretical prediction for fluids with large Prandtl number, due to Batchelor (1959). Even though air has a Prandtl number of 0[sdot ]7, the observations are in qualitative agreement with predictions of the theory. The non-dimensional wavenumber at which the increase in slope occurs is about 0[sdot ]02, in good agreement with observations in the ocean reported by Grant et al. (1968).For the two runs for which the stratification was stable, the normalized spectra of the temperature derivative fall on average slightly below the mean of the spectra of the remaining runs in the range in which the slope is approximately one-third. Hence the Reynolds number may not have always been sufficiently high to satisfy completely the conditions for an inertial subrange.Universal inertial-subrange constants were directly evaluated from one-dimensional dissipation spectra and found to be 0[sdot ]54 and 1[sdot ]00 for velocity and temperature, respectively. The constant for velocity is consistent with previously reported values, while the value for temperature differs from some of the previous direct estimates but is only 20% greater than the mean of the indirect estimates. This discrepancy may be explained by the neglect in the indirect estimates of the divergence terms in the conservation equation for the variance of temperature fluctuations. There is weak evidence that the one-dimensional constant, and hence the temperature spectra, may depend upon the turbulence Reynolds number, which varied from 1200 to 4300 in the observations reported.

Forced baroclinic ocean motions. II - The linear equatorial bounded case

Abstract: Results obtained in the first part of this paper (1976), which studied forced baroclinic ocean motions in the linear equatorial unbounded case, are extended to the case of an ocean bounded by two meridians. A complete solution for the linear spin-up in response to a switched-on, x-dependent wind stress is obtained in terms of five components: the unbounded nonoscillatory response to the wind stress forcing; the inertia-gravity waves generated when the forcing switches on, together with their reflections; the quasi-geostrophic Rossby modes forming the eastern boundary response and a western boundary layer; and the equatorial Kelvin wave component of the western boundary response. Kelvin waves are generated if the forcings include a zonal wind component that is symmetric about the equator or an antisymmetric meridional wind stress. For non-Kelvin symmetries, spin-up occurs entirely by the effects of Rossby waves emanating from the eastern boundary. The state approaches the steady solution as increasing numbers of these reach an interior point. In Kelvin symmetries, the Rossby waves act to bring the sea surface tilt to the steady value and the spin-up of the tilt proceeds as in non-Kelvin cases.