Defect Law Research Articles

Spectral derivation of the classic laws of wall-bounded turbulent flows.

We show that the classic laws of the mean-velocity profiles (MVPs) of wall-bounded turbulent flows—the ‘law of the wall,’ the ‘defect law’ and the ‘log law’—can be predicated on a sufficient condition with no manifest ties to the MVPs, namely that viscosity and finite turbulent domains have a depressive effect on the spectrum of turbulent energy. We also show that this sufficient condition is consistent with empirical data on the spectrum and may be deemed a general property of the energetics of wall turbulence. Our findings shed new light on the physical origin of the classic laws and their immediate offshoot, Prandtl’s theory of turbulent friction.

Defective Galton–Watson processes

ABSTRACTThe Galton–Watson process is a Markov chain modeling the population size of independently reproducing particles giving birth to k offspring with probability pk, k ⩾ 0. In this paper, we consider defective Galton–Watson processes having defective reproduction laws, so that ∑k ⩾ 0pk = 1 − ϵ for some ϵ ∈ (0, 1). In this setting, each particle may send the process to a graveyard state Δ with probability ϵ. Such a Markov chain, having an enhanced state space {0, 1, …}∪{Δ}, gets eventually absorbed either at 0 or at Δ. Assuming that the process has avoided absorption until the observation time t, we are interested in its trajectories as t → ∞ and ϵ → 0.

Industrial Law and the Productive Capacity of Labour in Nigeria

This Paper has discussed the various employment laws in the country and the inherent defects in them that have impacted negatively on staff productivity. The Paper came against the background of the low productive output of the Nigerian worker that has adversely affected the growth of the national economy and created room for fraud and corruption in the public service as well as the private sector. The Paper has found that aside of defective labour laws, the mentality of Nigerian courts to labour disputes, the negative attitude of Nigerian workers and poor work ethics, and the poor conditions of service in the labour sector all contribute to low output and productivity. Therefore, the Paper has suggested ways that these negative trends can be reversed among them, the amendment of the extant industrial laws and improvement of the working conditions of the Nigerian employee.

The Reynolds analogy and a new formulation of the temperature-defect law for a turbulent boundary layer on a plate

A rational asymptotic theory describing the dynamic and thermal turbulent boundary layer on a plate at zero pressure gradient is proposed. The fact that the flow depends on a finite number of governing parameters makes it possible to formulate algebraic closure conditions, which relate the turbulent shear stress and heat flux to mean velocity and temperature gradients. As a result of an exact asymptotic solution of the boundary-layer equations, the known laws of the wall for the velocity and temperature and the velocity and temperature defect laws as well as the expression for the skin-friction coefficient, the Stanton number, and the Reynolds-analogy factor are obtained. The latter implies two new formulations of the temperature-defect law, one of which is completely similar to the velocity-defect law and does not contain the Stanton number and the turbulent Prandtl number, and the other does not contain the skin-friction coefficient. A heat-transfer law that relates only thermal quantities is also obtained. The conclusions of the theory agree well with experimental data.

Effect of Freestream Motion on Heat Transfer Characteristics of Turbulent Offset Jet

The numerical simulation of turbulent offset jet flow has been carried out using k–ω shear stress transport (SST) model. The simulations have been done for the offset jet flow in the quiescent medium and also in the presence of an external stream. The effect of freestream velocity on the flow and heat transfer characteristics of turbulent offset jet has been reported. The offset ratio and Reynolds number of flow considered are 5.7 and 16,000, respectively. The presence of coflow stream has been found to reduce the entrainment of surrounding fluid into the jet which in turn reduces the heat transfer from the jet to the surrounding medium. The effect of freestream velocity on the important parameters like decay of the local maximum streamwise velocity, jet spread, reattachment length, velocity logarithmic profile, velocity defect law profile, decay of the local maximum streamwise temperature, variation of wall temperature, temperature similarity profile, and Nusselt number distribution has been discussed.

Turbulent boundary layer on a plate. Reynolds analogy and new formulations of the temperature defect law

AbstractA consistent asymptotic theory describing hydrodynamic and thermal turbulent boundary layers on a flat plate in zero pressure gradient is developed. The fact that the flow depends on a limited number of governing parameters allows us to formulate algebraic closure conditions that relate the turbulent shear stress and turbulent heat flux to mean velocity and temperature gradients. As a result of an exact asymptotic solution of the boundary‐layer equations, the known laws of the wall for the velocity and temperature and the velocity and temperature defect laws as well as the expressions for the skin‐friction coefficient, Stanton number, and Reynolds‐analogy factor are obtained. The latter implies two new formulations for the temperature defect law one of which is completely similar to the velocity defect law and does not contain the Stanton number and the turbulent Prandtl number, and the other does not contain the skin‐friction coefficient. A heat‐transfer law is obtained that relates only thermal quantities. The theoretical conclusions agree well with experimental data. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Experimental study of depth-limited open-channel flows over a gravel bed

Laboratory experiments of depth-limited open-channel flows over a gravel bed were conducted in the study. Two gravel patches with identical individual element size and different lengths (3.81m and 7.5m) were tested. The depth-limited uniform flow regime with relative submergence Sr (=D/ks) ranging from 2.68 to 5.94 was produced by adjusting the tailgate weir. The velocity profiles were measured by using both an ultra-sound velocity profiler (UVP) and an acoustic Doppler velocimeter (ADV). The conventional methods used to determine the zero-plane displacement and estimate the bed shear velocity were then reviewed and compared. The measured double-averaged (DA) velocity profiles were found to fit well with the log law and defect law with a non-universal Karman constant κ. κ-value remains nearly constant and in the range from 0.2 to 0.3 for the long patch (LP) cases and κ-values are scattered within a wider range from 0.3 to 0.5 for the short patch (SP) cases. While the Br-value in log law remains constant and equal to 8.5 for LP cases, the Br-value was found to decrease with the increase of the dimensionless roughness height ks+ for SP cases. The streamwise turbulence intensity distributions were found to be independent on the patch length and agree well with the available experimental data in the intermediate region and wall region. The Manning resistance coefficient and Darcy–Weisbach friction factor were analyzed. The κ-value decreases to 0.22 for the fitting of the logarithmic flow resistance law under small relative submergence. The value of the integration constant Ar in the logarithmic law falls within the normal range between 3.25 and 6.25.

New formulations of the temperature defect law for turbulent boundary layers on a plate

A consistent asymptotic theory describing hydrodynamic and thermal turbulent boundary layers on a flat plate in zero pressure gradient is developed. The fact that the flow depends on a limited number of governing parameters allows us to formulate algebraic closure conditions that relate the turbulent shear stress and turbulent heat flux to mean velocity and temperature gradients. As a result of an exact asymptotic solution of the boundary-layer equations, the known laws of the wall for the velocity and temperature and the velocity and temperature defect laws as well as the expressions for the skin-friction coefficient, Stanton number, and Reynolds-analogy factor are obtained. The latter implies two new formulations for the temperature defect law one of which is completely similar to the velocity defect law and does not contain the Stanton number and the turbulent Prandtl number, and the other does not contain the skin-friction coefficient. A heat-transfer law is obtained that relates only thermal quantities. The theoretical conclusions agree well with experimental data.

Spectral analogues of the law of the wall, the defect law and the log law

AbstractUnlike the classical scaling relations for the mean-velocity profiles of wall-bounded uniform turbulent flows (the law of the wall, the defect law and the log law), which are predicated solely on dimensional analysis and similarity assumptions, scaling relations for the turbulent-energy spectra have been informed by specific models of wall turbulence, notably the attached-eddy hypothesis. In this paper, we use dimensional analysis and similarity assumptions to derive three scaling relations for the turbulent-energy spectra, namely the spectral analogues of the law of the wall, the defect law and the log law. By design, each spectral analogue applies in the same spatial domain as the attendant scaling relation for the mean-velocity profiles: the spectral analogue of the law of the wall in the inner layer, the spectral analogue of the defect law in the outer layer and the spectral analogue of the log law in the overlap layer. In addition, as we are able to show without invoking any model of wall turbulence, each spectral analogue applies in a specific spectral domain (the spectral analogue of the law of the wall in the high-wavenumber spectral domain, where viscosity is active, the spectral analogue of the defect law in the low-wavenumber spectral domain, where viscosity is negligible, and the spectral analogue of the log law in a transitional intermediate-wavenumber spectral domain, which may become sizable only at ultra-high$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}_{\tau }$), with the implication that there exist model-independent one-to-one links between the spatial domains and the spectral domains. We test the spectral analogues using experimental and computational data on pipe flow and channel flow.

평판 위 흐름 Defect Law 영역의 난류 특성 연구

To investigate the turbulence characteristics within the boundary layer over a flat plate, an experimental study was performed using a PIV technique in a circular water channel. For two water velocities, 0.92 and 1.99 m/s, the water velocity profiles were taken and analyzed to determine turbulent characteristics such as the Reynolds stress, Taylor micro-length scale, and Kolmogorov length scale within the defect law region of the boundary layer. These analysis methods may be applied to research on the friction drag reduction technology using micro-bubbles or an air sheet over the surface of a ship's hull, because the physical reason for the friction drag reduction could be found by understanding the variation of the turbulence characteristics and structures in the boundary layer.

Numerical study of turbulent wall jet over multiple-inclined flat surface

In the present work flow field of a turbulent oblique wall jet with multiple flat inclined wall has been numerically investigated. The Reynolds averaged Navier–Stokes (RANS) equations for incompressible flow are solved in transformed geometry using boundary-fitted coordinate system. The aim of the work is to investigate the influence of three inlet conditions: uniform, parabolic and trapezoidal inlet velocity profiles, on the mean flow and turbulent characteristics of the jet flowing tangentially along a multiple inclined solid wall. The results show that the inlet conditions affect the flow in the vicinity of the nozzle exit area and also the far field region. Mean flow has been presented in terms of evolution of mean velocity profiles, jet spread, decay of streamwise maximum velocity, distribution of wall pressure and skin friction coefficients, and streamwise variations of integrated momentum and energy fluxes. Discontinuities in the boundary slope of the segmented wall produces oscillations in distribution of skin friction and pressure coefficients. Reduced pressure in the region of wall discontinuity appeared to be causing the deflected jet to remain attached to the wall. The growth of the inner layer of the jet is relatively high with uniform inlet velocity profiles in the near-field region, but in the far-field region the inner layer grows with higher rate for the trapezoidal and the parabolic profiles. The rate of decay of maximum streamwise velocity for oblique wall jet on segmented wall is higher than that of plane wall jet. The jet with parabolic and trapezoidal inlet profiles show higher rate of decay than that with a uniform inlet profile. Similarity of streamwise velocity and velocity component parallel to the oblique wall has been observed in the developed region of flow. The volume flow rate of the jet varies linearly along the flow direction due to entrainment of the fluid from the surrounding. Sudden change in slope of the wall causes a sharp rise in entrained volume of fluid. The distribution of near wall velocity in the inner coordinates of the boundary layer is grossly altered and the profiles do not conform to the universal velocity profiles in the form of the law of the wall. Relatively higher position of defect law data signifies comparatively lower turbulence in the outer mixing layer. Cross-stream distribution of Reynolds stresses shows that the effect of inlet velocity profiles is more pronounced in the near field region of flow. Self-similarity is observed in the normalized turbulent shear stress profiles whereas considerable scatter can be noticed in the profiles of turbulent normal stresses for all the inlet velocity conditions. Two distinct peaks in the normalized turbulent kinetic energy profiles characterize the two shear layers of the jet and shifting of the outer peak toward the wall suggests a strong interaction of the outer layer with the inner layer as the jet develops.

Numerical characterization of a planar turbulent offset jet over an oblique wall

In the present study, mean velocity and turbulence characteristics of an oblique offset jet with an offset ratio of 7.0 and oblique angle, α=10° have been numerically investigated in details. A comparative study of mean flow parameters, and turbulence stresses along the jet centerline has also been performed to investigate the effect of variation of oblique angle. The Reynolds number considered for the present study is 15000 for which the flow field becomes fully turbulent. Flow field has been represented in terms of streamwise and lateral velocity contours, static pressure contour, and streamwise velocity and static pressure profiles at different locations along the oblique offset plate. Distribution of Reynolds stresses in terms of spanwise, lateral and turbulent shear stresses, and turbulent kinetic energy and its dissipation rate have been presented to describe the turbulent characteristics. Spanwise integral of mean momentum, static pressure, Reynolds stress and energy have been computed along the downstream direction. Similarity of streamwise velocity and velocity component parallel to the oblique offset plate has been observed in the wall jet region of the reattached flow. The jet growth rate has been presented in terms of half-width, the decay of maximum velocity, variation of maximum velocity location and growth of length scale. Effect of variation of oblique angles on the streamwise variation of static pressure and skin friction coefficient have been compared for different angles. In the wall jet region streamwise variation of centerline turbulent intensities shows increased values with increase in oblique angle while turbulence intensities along the jet centerline in the preimpingement region remain unaffected by change in oblique angles. Near-wall velocity profiles have been presented by logarithmic law profiles and velocity defect law. Velocity distribution of oblique offset jet in inner coordinates shows differences with that of the boundary layer as well as plane wall jet.

Use of the velocity defect law to assess sand transport rates

Abreu, T., Silva, P.A., Sancho, F. and Michallet, H., 2013. Use of the velocity defect law to assess sand transport rates.This work uses a simple method based on the defect law that is able to reproduce the horizontal velocities within the wave bottom boundary layer, using a limited number of parameters. The theory is checked against the measured velocity profiles gathered with a high resolution Acoustic Doppler Velocity Profiler. The velocities were collected during an experimental project performed at the Large Oscillating Water Tunnel of Deltares under flat-bed/sheet flow conditions, allowing the analysis of the effects of wave nonlinearities and of a net current on the sediment transport processes. The velocity estimates are used to infer the time-varying bed shear stress, using the momentum-integral method. Secondly, the bed shear stresses are incorporated in a quasi-steady bed load formulation and the transport rate predictions are compared with the net transport rate measurements obtained in the same facility. The results show that the methodology presented in this work provides a suitable estimation of the horizontal near-bed sediment transport rates under non-linear oscillatory flows.

Effects of a fully submerged boulder within a boulder array on the mean and turbulent flow fields: Implications to bedload transport

The objective of this coupled experimental and numerical study is to provide insight into the mean and turbulent flow fields within an array of fully submerged, isolated, immobile boulders. Our study showed that the velocity defect law performed well for describing the mean flow around the boulder within the array. A prerequisite, however, was to accurately estimate the spatial variability of u* around the boulder, which was achieved via the boundary characteristics method. The u* exhibited considerable spatial variability within the array and form roughness was shown to be up to 2 times larger than the skin roughness in the boulder near-wake region. Because the boulders bear a significant amount of the flow shear, the available bed shear stress for entrainment of the mobile sediment, τ ols , near the boulders was roughly 50% lower than the ambient τ ols . The τ ols variability induced by the boulders could lead to a threefold overestimation of the sediment transport rate.

Effects of accelerating and decelerating flows in a channel with vegetated banks and gravel bed

Experiments were conducted in an 8 m long, 40 cm wide, and 60 cm deep re-circulating flume with vegetated banks and gravel bed to study the effects of accelerating and decelerating flows on the flow structure. Significant dip phenomenon was observed in velocity profiles under decelerating flow and the velocity defect law was not suitable due to the effect of vegetated banks. Near the vegetated banks, the turbulence intensities illustrated convex distribution for both accelerating and decelerating flows. Application of quadrant analysis revealed that the effect of vegetated banks is important on the flow structure and the Reynolds stress distribution. Accelerating and decelerating flows did not show a significant difference on percentage of time occupied by each kind of turbulent event within bursting cycle, but affected the shape of joint probability distributions of turbulent intensities in the horizontal and vertical directions.

Deviations from the law of velocity defect observed for small reynolds numbers

Deviations from the velocity defect law for a turbulent flow through plane channels and tubes with a circular cross sections are studied by direct numerical simulation based on the results obtained by different authors.

VELOCITY DEFECT LAW IN THE WAVE BOTTOM BOUNDARY LAYER

This work presents a simple method based on the defect law (Nielsen,1992) to reproduce the velocity vertical profile within the wave bottom boundary layer. The model results agree fairly well with the ADVP measurements and show that the defect law reproduces typical features of the oscillatory boundary layer: the velocity magnitude first increases with distance from the bed, with an overshoot at approximately 3 cm above the bed. There is a phase shift in the velocity that is maximum at about 1cm above the bed. Processing of the bed shear stress as well as velocities estimates within the sheet flow layer is under progress. (e.g. Ruessink et al., submitted)

Defective Law

Defective Law

Turbulent flow and organized motions over a two-dimensional rough wall

Particle image velocimetry measurements at relatively low Reynolds number in a turbulent boundary layer over a two-dimensional roughened surface are presented and compared to those for a smooth wall. The main goal of the study was to give a contribution to the understanding of the behavior of the turbulent organized motions, when the influence of the wall roughness on the turbulent structures may be expected to be noticeable, i.e., for relatively high roughness and relatively low Reynolds number. In addition, the flow behavior with respect to the wall similarity hypothesis in the case of moderately large values of the roughness height k and moderately low Reynolds number was tested. The behavior of the turbulent coherent motions appears to be similar for the two flows in the outer layer. Quantitative differences, observed mainly in the roughness sublayer, are enlightened comparing spatial velocity correlations and vortex population density and strength in the wall-normal-streamwise plane and in planes parallel to the wall. Mean flows expressed in velocity defect law, inner scaled streamwise Reynolds normal stress, and Reynolds shear stresses profiles show excellent agreement in the outer layer between the rough wall and the smooth surface. Wall normal Reynolds normal stress profiles, instead, show some small discrepancy, but within the measurement uncertainty.

Trade, population growth, and the environment in developing countries

We examine pollution in a developing country where fertility is endogenous and wealth increases welfare through status. When the country has defective environmental laws, it has a comparative advantage in capital-intensive “dirty” goods. Gains from trade due to trade liberalization then increase income and boost population growth. With strong incentives to save, they also stimulate investment, which hampers population growth. Because population growth crowds out labor supply, production of capital-intensive dirty goods first increases and then decreases. This yields a typical environmental Kuznets path: pollution increases at the earlier stages but decreases at the later stages of development.