Boundary Layer Method Research Articles

Laminar flow of power-law fluids in the entrance region of a pipe

The study is concerned with developing laminar flow of a power-law fluid in a circular tube. The analysis extends the filled-region concept, used previously to study Newtonian fluid flow, to the more general class of power-law fluids. Flow is analyzed in both the inlet and filled regions using an integral boundary layer method. Results obtained provide the lengths of the entrance, inlet and filled regions as a function of the generalized Reynolds number and the power-law fluid index. In addition, the variations of the local friction factor, the pressure drop and the centerline velocity along the axial coordinate are also provided. The available models are compared with the present one on the basis of experimental data. The present results are found to reach asymptotically the fully developed values, and also to be in good agreement with all available experimental data.

Measuring nighttime CO 2 flux over terrestrial ecosystems using eddy covariance and nocturnal boundary layer methods

The respiration from plants and soil is an important component of the carbon balance of ecosystems. Its measurement is challenging due to the relatively small size of the carbon dioxide fluxes and because these fluxes occur under environmental conditions that are frequently unfavorable for flux measurements. Micrometeorological techniques based on turbulent transfer frequently underestimate CO 2 fluxes during nighttime conditions. An approach based on calculating the CO 2 budget in the nocturnal boundary layer (NBL) might be an alternative during light wind conditions to estimate nighttime CO 2 fluxes. This study presents typical net CO 2 efflux observations measured at night over agricultural crops for several years near Ottawa, Ont. and over an old black spruce stand near Candle Lake, Sask. during the intensive field campaigns of the Boreal Ecosystem Atmosphere Study (BOREAS). We used the eddy covariance technique for windy nights and the NBL budget approach for calm nights. Criteria for screening data into either windy or calm conditions were made using the friction velocity ( u ∗) and the standard deviation of the vertical wind speed ( σ W). The threshold at which the 30 min turbulent CO 2 flux observations were independent of u ∗ or σ W and had limited scatter were determined to be in the range 0.075–0.1 m s −1 for u ∗ and σ W, and about 1.5 m s −1 for horizontal wind speed ( U) for multiple years of corn and soybean data. σ W≥0.4 m s −1 was shown to be a good screening threshold over the black spruce canopy. Current methods for handling nocturnal CO 2 data involve systematically replacing data during calm conditions where eddy covariance is deficient with those from windy conditions. This can lead to an overestimation of the nocturnal CO 2 flux. We suggest a variation in this procedure which, through the screening of entire nights, allows the retaining of an acceptable proportion of calm periods within predominantly windy nights. The NBL budget method requires calm nights for measuring the respiration unless other budget terms are quantified. A good agreement was found between CO 2 flux measured using the NBL approach when the NBL was well developed and the eddy covariance technique, when restricted to windy nights. The NBL profiles integrate a larger area than eddy covariance, which means that high-emission spots can be included with this kind of approach.

Boundary-layer isotope dilution/mass balance methods for measurement of nocturnal methane emissions from grazing sheep

Following advances with methods for 13C/ 12C isotopic analysis of methane in small (⩽4 L) air samples, new isotope dilution techniques are proposed for measurement of methane emissions at the paddock scale from grazing ruminant animals. These techniques combine measurement of the isotopic δ 13CH 4 composition of air samples with a non-intrusive mass balance method applied in the nocturnal boundary layer. Flux estimates from trials using the isotope dilution techniques are compared with estimates based on scaling up individual animal emission measurements using a rumen gas tracer technique. The methane flux assessed by the latter technique ranged from 35 to 70 mg (CH 4) m −2 d −1 with a stocking density between 10 and 20 sheep ha −1. The isotope dilution based nocturnal boundary-layer estimates generally agreed to better than a factor of 2 and usually to within 20% of the average of individual animal emission rate per unit area of paddock. Both static and advecting mass balance methods are developed. In the advecting case, the upwind/downwind contrast in δ 13C was typically 0.2–0.5‰. Care was necessary with air sampling to avoid error in this small contrast contributing to error in the flux. Agreement between concentration- and isotope-based nocturnal boundary layer methods and the sheep breath measurements indicated that sample representativeness was generally good. Factors which affect the accuracy of the method are examined and include variability in nocturnal mixing height, the assumed δ 13CH 4 composition of the source sheep breath and diurnal patterns in sheep emission. This paper establishes new techniques useful in the paddock to landscape scale although widespread application awaits further development of technology for rapid and repeatable field analysis of δ 13CH 4 in small samples.

Skin effect in massive conductors used in pulsed electrical devices: I. Electromagnetic field of massive conductors

The diffusion of a pulsed electromagnetic field into massive conductors with an arbitrary smooth surface is considered for the case where the field penetration depth is small. By using the boundary layer method, an asymptotic solution for the electromagnetic field is constructed. First-and second-order corrections to the limiting solution, which corresponds to the field distribution at an indefinitely high conductivity of the conductors, are found. Time dependences of the first-and second-order approximations to the electric field on the surface of the conductor are determined.

Fluid Dynamics: Theory, Computation, and Numerical Simulation

Fluid Dynamics: Theory, Computation, and Numerical Simulation

Air–land exchanges of CO 2, CH 4 and N 2O measured by FTIR spectrometry and micrometeorological techniques

Micrometeorological flux-gradient and nocturnal boundary layer methods were combined with Fourier transform infrared (FTIR) spectroscopy for high-precision trace gas analysis to measure fluxes of the trace gases CO 2, CH 4 and N 2O between agricultural fields and the atmosphere. The FTIR measurements were fully automated and routinely obtained a precision of 0.1–0.2% for several weeks during a measurement campaign in October 1995. In flux-gradient measurements, vertical profiles of the trace gases were measured every 30 min from the ground to 22 m. When combined with independent micrometeorological measurements of water vapour fluxes, trace gas fluxes from the underlying surface could be determined. In the nocturnal boundary layer method the rate of change in mass storage in the 0–22 m layer was combined with fluxes measured at 22 m to estimate surface fluxes. Daytime fluxes for CO 2 were −0.78±0.40 (1 σ) mg CO 2 m −2 s −1. Daytime fluxes of N 2O and CH 4 were very small and difficult to measure reliably using the flux-gradient technique, despite the high precision of the concentration measurements. Mean daytime flux for N 2O was 17±48 ng N m −2 s −1, while the corresponding flux for CH 4 was 47±410 ng CH 4 m −2 s −1. The mean nighttime flux of CO 2 estimated using the nocturnal boundary layer method was +0.15±0.05 mg CO 2 m −2 s −1, in good agreement with chamber measurements of respiration rates. Nighttime fluxes of CH 4 and N 2O from the nocturnal boundary layer method were 109±69 ng CH 4 m −2 s −1 and 2±3.2 ng N m −2 s −1, respectively, in good agreement with chamber measurements and inventory estimates based on the sheep and cattle stocking rates in the region. The suitability of FTIR-based methods for long term monitoring of spatially and temporally averaged flux measurements is discussed.

SLOW PERIODIC MOTIONS WITH INTERNAL SLIDING MODES IN VARIABLE STRUCTURE SYSTEMS

Singularly perturbed relay systems (SPRS) in which the reduced systems have the stable periodic motions with internal sliding modes are studied. The slow motion integral manifold of such systems consists of the parts which correspond to the different values of relay control and the solutions may contain the jumps from one part of the slow manifold to another. For such systems a theorem about existence and stability of the periodic solutions is proved. An algorithm of asymptotic representation for this periodic solutions using boundary layer method is presented. It is shown that in the neighbourhood of the break away point the asymptotic representation starts with the first order boundary layer function.

Slow periodic motions in variable structure systems

Singularly perturbed relay control systems (SPRCS) with stable periodic motion in reduced systems are studied here. The slow motions integral manifold of such systems consists of parts that correspond to different values of control and the solutions of SPRCS contain the jumps from one part of the slow manifold to the other. The theorems about existence and stability of the slow periodic solutions are proved. An algorithm of asymptotic representation for this periodic solutions using a boundary layer method is suggested.

Slow periodic motions with internal sliding modes in variable structure systems

Singularly perturbed relay systems (SPRS) in which the reduced systems have the stable periodic motions with internal sliding modes are studied. The slow motion integral manifold of such systems consists of the parts which correspond to the different values of relay control and the solutions may contain the jumps from one part of the slow manifold to another. For such systems a theorem about existence and stability of the periodic solutions is proved. An algorithm of asymptotic representation for this periodic solutions using boundary layer method is presented. It is proved that in the neighbourhood of the break away point the asymptotic representation starts with the first order boundary layer function.

Airfoil Validation Using Coupled Navier-Stokes and e Transition Prediction Methods

Navier-Stokes airfoil computations coupled to e* transition prediction are feasible, provided the limiting N factor is known beforehand. In the present study a procedure is outlined to validate free-transition airfoil experiments in wind tunnels for which the limiting N factor is not known a priori. The approach does not rely on mesh adaption procedures to obtain adequate laminar viscous layer data from Navier-Stokes computations for the stability analysis. To the contrary, the laminar viscous layer is computed by a boundary-layer method applying as input the pressure distribution from Navier-Stokes computations on initial meshes. Two measurement campaigns are validated: the NLF(1)-0416 laminar airfoil in the low-speed NASA Langley Low-Turbulence Pressure Tunnel LTPT and the NACA 64*A015 airfoil in the NASA Ames 12-Foot Pressure Tunnel. The free-transition measurements in both tunnels include pressure distributions and transition locations and, in supplement, for the NLF(1)-0416 laminar airfoil lift and drag measurements. The computational results document the validity of the present approach, the existence of a constant limiting N factor for a specific wind tunnel, and an excellent agreement with the experimental findings.

The quantum mechanical two-Coulomb-centre problem inthe Dirac equation framework

The asymptotic expansions (at small and large internuclear distancesR) of the eigenvalues (potential curves) E(R) of thetwo-Coulomb-centre problem are obtained. The Dirac equation with anaxially symmetrical potential, not allowing complete separation ofvariables, is solved analytically by the Wentzel-Kramers-Brillouinapproach and boundary-layer method. In the framework of thisscheme, the relativistic two-Coulomb-centre wavefunction isconstructed. The first two terms of the asymptotic (at largeinternuclear distance) behaviour of the exchange interactionpotential for an ion with an atom are calculated.

Unsteady RANS method for surface ship boundary layer and wake and wave field

AbstractResults are reported of an unsteady Reynolds‐averaged Navier–Stokes (RANS) method for simulation of the boundary layer and wake and wave field for a surface ship advancing in regular head waves, but restrained from body motions. Second‐order finite differences are used for both spatial and temporal discretization and a Poisson equation projection method is used for velocity–pressure coupling. The exact kinematic free‐surface boundary condition is solved for the free‐surface elevation using a body‐fitted/free‐surface conforming grid updated in each time step. The simulations are for the model problem of a Wigley hull advancing in calm water and in regular head waves. Verification and validation procedures are followed, which include careful consideration of both simulation and experimental uncertainties. The steady flow results are comparable to other steady RANS methods in predicting resistance, boundary layer and wake, and free‐surface effects. The unsteady flow results cover a wide range of Froude number, wavelength, and amplitude for which first harmonic amplitude and phase force and moment experimental data are available for validation along with frequency domain, linear potential flow results for comparisons. The present results, which include the effects of turbulent flow and non‐linear interactions, are in good agreement with the data and overall show better capability than the potential flow results. The physics of the unsteady boundary layer and wake and wave field response are explained with regard to frequency of encounter and seakeeping theory. The results of the present study suggest applicability for additional complexities such as practical ship geometry, ship motion, and maneuvering in arbitrary ambient waves. Copyright © 2001 John Wiley & Sons, Ltd.

Creeping waves on a highly elongated body of revolution

Creeping waves play an important role in diffraction by a smooth convex body and give an asymptotic of the diffracted field in the shadow. Known results obtained by the boundary-layer method do not allow us to explain some of the properties of creeping waves on highly elongated bodies. In this paper, creeping waves on highly elongated bodies are studied in the case where the binormal curvature of the surface is asymptotically large. The asymptotics derived contains solutions of a differential equation of the Heun type. The analysis of the dispersion equation for the surface waves is carried out numerically. It is discovered that the magnetic creeping wave travels along the surface of a highly elongated body with much less attenuation than predicated by the usual theory. Bibliography: 5 titles.

A ‘Boundary Layer’ Method for Obtaining an Approximate Equation for the Velocity–Time History of an Accelerating Sphere

A sphere starting from rest in a stationary fluid accelerates under the action of gravity and eventually reaches a terminal velocity. The descriptive equation for the acceleration of the sphere is a first order, non-linear, ordinary differential equation, which can be solved using a finite difference scheme. In this study, we present a method (obtained from boundary layer methods) of generating a velocity—time equation to obtain an alternative solution. The method is used to solve a specific example and the results are compared to those obtained with the numerical solution. The comparison yields a small error, and so the polynomial approximation method can be used as an alternative technique to describing the motion of an accelerating sphere. The accelerating sphere problem is usually encountered in a study of flow past immersed objects. Boundary layer methods are usually taught at the graduate level. The advantage of the method presented here is that boundary layer methods can be introduced to undergraduates in a first or second course in fluid mechanics.

Estimation of Meteorological Parameters for Air Quality Management: Coupling of Sodar Data with Simple Numerical Models

Abstract In this paper an attempt is made to couple sodar data and simple numerical models to calculate the wind field and the boundary layer parameters that are relevant to air quality monitoring and studies. For this purpose, a diagnostic, mass-consistent wind field model and a mixed-layer model are used. The sodar data used in the analysis were taken from three Doppler sodars that were operating in the area of Rome, Italy. A similarity method for the convective boundary layer is used to calculate the surface fluxes of heat and momentum from sodar measurements of the mean horizontal wind and the vertical velocity variance. The calculated surface fluxes of heat and momentum are used as input data to the mixed-layer model. The day chosen for the analysis is 18 July 1992, on which a sea-breeze circulation developed in the area. The main characteristics of the wind field and the boundary layer structure are presented and discussed.

Improvement of one-equation boundary-layer integral method for pool film-boiling heat transfer from a downward-facing surface

Improvement of one-equation boundary-layer integral method for pool film-boiling heat transfer from a downward-facing surface

Boundary-Layer Transition, Separation, and Reattachment on an Oscillating Airfoil

integral value obtained with this method results very close to that obtained with the potential/boundary-layer simulation. However, it seems that the better prediction of the local behavior of the friction drag is obtained with the Reynolds-stress closure method. Note that for the RANS simulations the values of c f obtained on both the upper and lower airfoil surfaces are reported. Because symmetry is not perfect, because of discretization errors, two slightly different lines can be distinguished. InFig.1b,thelocalfrictiondragcoefe cientdistributionsobtained by the RANS and the boundary-layer methods are compared to the theoretical results for the e at plate. In both cases, the friction drag on the proe le is higher than that on the e at plate in the leading-edge zone, although it is lower near the trailing edge. This behavior is consistent with the effects of the chordwise pressure gradient. Because the pressure distributions are practically the same in all of the simulations, the Reynolds-stress closure method predicts a larger variation of the local coefe cient c f with the pressure gradient than the potential/boundary-layer simulation. However, it is expected that RANS simulations give a better representation of the effects of the pressure gradient than the boundary-layer method; thus, it is not clear which solution is the most accurate in the leading-edge region. The computations were carried out on a Pentium III 500-MHz XION processor, with 512 MB RAM. The computing time for the case with 34,000 total cells was about 70 min for the standard k‐e closure method, 110 min for the RNG k‐e closure method, and 150 min for the Reynolds-stress closure method (with a few seconds for the potential/boundary-layer simulations ). Therefore, the Reynolds-stress closure method appears signie cantly more time consuming. In general, the RANS calculations seem to require computational resources, both memory and computing time, which would become prohibitive in three-dimensional calculations. Conclusions The capabilities of a solver of the RANS equations in predicting the friction drag overanairfoil havebeen investigatedthrough comparisonwith the valuesgiven by a coupled potential/boundary-layer method, for different Reynolds numbers. Preliminarily, the near-wall grid resolution required to obtain the grid independence ofthe friction drag in the RANScalculations has been assessed. It appears that, for all of the considered Reynolds numbers, a large amount of computational points is required, which would lead to an unaffordable mesh size in three-dimensional simulations. Even on these highly ree ned grids, the value of the global CF is overestimated by all of the turbulence models because they are not able to predict the boundary-layer transition. If comparison is made with the value given by the potential code coupled with a fully turbulent boundary layer, satisfactory agreement is obtained with the RNG k‐e and the Reynolds-stress closure models. The best global agreement is given by the RNG k‐e model. However, from the analysis of the chord distribution of the local c f , it appears that this is due to compensation between an overestimate near the leading edge and an underestimation at the trailing edge. The best local agreement is obtained, as expected, with the Reynolds-stress model; the only signie cant discrepancy with the BLOWS results is a less steep decrease of the c f near the leading edge. Because the pressure distribution is almost identical, it appears that the RANS simulation with this closure models predicts larger variations of the friction coefe cient with the pressure gradient.Becausethe boundary-layersolversare notwellsuitedfor e ows with high-pressure gradients, it is not clear whether the value of c f obtained by potential/boundary-layer simulation is indeed more accurate in the region near the leading edge. Finally, the RANS simulations require in general large computational time, and this increases signie cantly with the accuracy of the turbulence closure model. Thus, this analysis indicates that an accurate prediction of the friction drag around complex aeronautical cone gurations by RANSmethods remainsanextremelydife cult task with the present computer capabilities.

Mass transport in a partially covered fluid-filled cavity

A method of computing the concentration field of dissolved material inside an etch-hole is presented. With a given velocity field, approximate convection–diffusion equations are formulated using a number of assumptions, and analytical descriptions for the concentration in different parts of the domain are obtained. By coupling these descriptions the concentration field can be computed. The assumptions and the results are validated by comparison with solutions based on a finite-volume method. Results of the boundary-layer method are given for two characteristic etch-hole geometries. The described boundary-layer method is efficient in terms of computational time and memory, because it does not require the construction of a computational grid in the interior of the domain. This advantage will be exploited in a future paper where the method will be used to simulate wet-chemical etching.

Ray solution of a backward–forward parabolic problem for data-handling systems

A problem for a backward–forward parabolic partial differential equation with a small diffusion coefficient ε is analyzed. The problem arises in describing the stochastic behaviour of a data-handling system with a large number of sources which turn on or off at random. We solve the problem asymptotically for ε small by using the ray method and the boundary layer method.

Periodic Unsteady Flow Aerodynamics and Heat Transfer: Studies on a Curved Surface, Combined Part I and II

Aerodynamic and heat transfer investigations were done on a constant curvature curved plate in a subsonic wind tunnel facility for various wake passing frequencies and zero pressure gradient conditions. Steady and unsteady boundary layer transition measurements were taken on the concave surface of the curved plate at different wake passing frequencies where a rotating squirrel‐cage generated the unsteady wake flow. The data were analyzed using timeaveraged and ensemble‐averaged techniques to provide insight into the growth of the boundary layer and transition. Ensemble‐averaged turbulence intensity contours in the temporal spatial domain showed that transition was induced for increasing wake passing frequency and structure. The local heat transfer coefficient distribution for the concave and convex surface was determined at those wake passing frequencies using a liquid crystal heat transfer measurement technique. Detailed aerodynamic and heat transfer investigations showed that higher wake passing frequency caused transition to occur earlier on the concave surface. Local Stanton numbers were also calculated on the concave surface and compared with Stanton numbers predicted using a differential boundary layer and heat transfer calculation method. On the convex side, no effect of wake passing frequency on heat transfer was observed due to a separation bubble that induced transition.