Velocity Defect Law Research Articles

A velocity-defect law for a self-similar turbulent boundary layer with pressure gradient

A velocity-defect law for a self-similar turbulent boundary layer with pressure gradient

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.

Asymptotic study of turbulent Poiseuille flow with wall transpiration

AbstractAn incompressible, pressure–driven, fully developed turbulent flow between two parallel walls, with an extra constant transverse velocity component, is considered. A closure condition is formulated, which relates the shear stress to the first and second derivatives of the longitudinal mean velocity. The closure condition is derived without invoking any special hypotheses on the nature of turbulent motion, only taking advantage of the fact that the flow depends on a finite number of governing parameters. By virtue of the closure condition, the momentum equation is reduced to the boundary–value problem for a second–order differential equation, which is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction velocity. A limiting transpiration velocity is obtained, such that the shear stress at the injection wall vanishes, while the maximum point on the velocity profile approaches the suction wall. In this case, a sublayer near the suction wall appears where the mean velocity is proportional to the square root of the distance from the wall. A friction law for Poiseuille flow with transpiration is found, which makes it possible to describe the relation between the wall shear stress, the Reynolds number, and the transpiration velocity by a function of one variable. A velocity defect law, which generalizes the classical law for the core region in a channel with impermeable walls to the case of transpiration, is also established. In similarity variables, the mean velocity profiles across the whole channel width outside viscous sublayers can be described by a one–parameter family of curves. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Analytical study of turbulent Poiseuille flow with wall transpiration

An incompressible, pressure-driven, fully developed turbulent flow between two parallel walls, with an extra constant transverse velocity component, is considered. A closure condition is formulated, which relates the shear stress to the first and the second derivatives of the longitudinal mean velocity. The closure condition is derived without invoking any special hypotheses on the nature of turbulent motion, only taking advantage of the fact that the flow depends on a finite number of governing parameters. By virtue of the closure condition, the momentum equation is reduced to the boundary-value problem for a second-order differential equation, which is solved by the method of matched asymptotic expansions at high values of the logarithm of the Reynolds number based on the friction velocity. There are three characteristic flow regions in the channel: the core region and two wall regions near injection and suction walls. For each region, the solution is constructed. The asymptotic matching gives formulas for the wall shear stress and the maximum mean velocity. A limit transpiration velocity is obtained, such that the shear stress at the injection wall vanishes, while the maximum point on the velocity profile approaches the suction wall. In this case, a sublayer near the suction wall appears where the mean velocity is proportional to the square root of the distance from the wall. A friction law for Poiseuille flow with transpiration is found, which makes it possible to describe the relation between the wall shear stress, the Reynolds number, and the transpiration velocity by a function of one variable. A velocity defect law, which generalizes the classical law for the core region in a channel with impermeable walls to the case of transpiration, is also established. In similarity variables, the mean velocity profiles across the whole channel width outside viscous sublayers can be described by a one-parameter family of curves. The theoretical results obtained are in good agreement with available direct numerical simulation data.

Investigations of the law-of-the-wall over sparse roughness elements

[1] This paper examines the application of the law-of-the-wall or gradient method for calculating the shear velocity, roughness length, and displacement height over three increasing roughness densities replicated with three different sized cubes within a recirculating wind tunnel. We compare these aerodynamic parameter estimates with estimates of the same parameters derived from other established methods: Reynolds stress analysis and the outer-layer velocity-defect law. By using more than one roughness height for the same roughness density (λ), dependencies of these parameters on roughness element height were also evaluated. Using the vertical wind speed logarithmic profile layer (determined graphically), resulted in shear velocity estimates that are greater by more than a factor of two than those determined using hot-film anemometry. The law-of-the-wall method provided a good estimate of the roughness length when applied to only that portion of the wind speed profile identified by Reynolds stress measurements to be within the constant stress layer; however, the shear velocity was overestimated by an average of 43% compared with that measured directly by hot-film anemometry. The best prediction of both of the roughness length and shear velocity, compared to estimates using Reynolds stress analysis, was obtained using the outer-layer velocity-defect law. We advocate that the velocity-defect law method be used in wind tunnel testing for calculating the shear velocity and roughness length from velocity profiles over sparsely spaced roughness elements, or when flow is highly heterogeneous, instead of the law-of-the-wall.

Analysis of power law and log law velocity profiles in the overlap region of a turbulent wall jet

The two-dimensional turbulent wall jet on a flat surface without free stream is analysed at a large Reynolds number, using the method of matched asymptotic expansions. The open mean equations of the turbulent boundary layer are analysed in the wall and wake layers by the method of matched asymptotic expansions and the results are matched by the Izakson–Millikan–Kolmogorov hypothesis. In the overlap region, the outer wake layer is governed by the velocity defect law (based on U m , the maximum velocity) and the inner layer by the law of the wall. It is shown that the overlap region possesses a non-unique solution, where the power law region simultaneously exists along with the log law region. Analysis of the power law and log law solutions in the overlap region leads to self-consistent relations, where the power law index, α , is of the order of the non-dimensional friction velocity and the power law multiplication constant, C , is of the order of the inverse of the non-dimensional friction velocity. The lowest order wake layer equation has been closed with generalized gradient transport model and a closed form solution is obtained. A comparison of the theory with experimental data is presented.

Incompatibility of the exponential scaling law for a zero pressure gradient boundary layer flow with Reynolds averaged turbulence models

Analyzing the multipoint-correlation equations for parallel turbulent shear flows and zero pressure gradient turbulent boundary layer flows with the help of Lie symmetry methods, a new exponential scaling law has been derived in the work by Oberlack [“A unified approach for symmetries in plane parallel turbulent shear flows,” J. Fluid Mech. 427, 299 (2001)]. In the frame work of Lie group methods these solutions are called invariant solutions. From experiments [B. Lindgren, J. M. Österlund, and A. Johansson, “Evaluation of scaling laws derived from Lie group symmetry methods in zero-pressuere-gradient turbulent boundary layers,” J. Fluid Mech. 502, 127 (2004)] and direct numerical simulations [G. Khujadze and M. Oberlack, “DNS and scaling laws from new symmetry groups of ZPG turbulent boundary layer flow,” Theor. Comput. Fluid Dyn. 18, 5 (2004)] the exponential velocity profile was clearly validated in the mid-wake region of high Reynolds number flat-plate boundary layers. It was identified as an explicit analytic form of the velocity defect law. Implementing the latter invariant solution into various Reynolds stress models it was found that none of the investigated models are in accordance with the exponential velocity law.

Evaluation of scaling laws derived from Lie group symmetry methods in zero-pressure-gradient turbulent boundary layers

New scaling laws for turbulent boundary layers recently derived (see Oberlack 2000) using Lie group symmetry methods have been tested against experimental data from the KTH database for zero-pressure-gradient turbulent boundary layers. The most significant new law predicts an exponential variation of the mean velocity defect in the outer (wake) region. It was shown to fit the experimental data very well over a large part of the boundary layer, from the outer part of the overlap region to about half the boundary layer thickness ( $\delta_{99}$ ). In the outermost part of the boundary layer the velocity defect falls more rapidly than predicted by the exponential law. This can partly be attributed to intermittency in that region but the main cause stems from non-parallel effects that are not accounted for in the derivation of the exponential law. The two-point correlation function behaviour in the outer region, where an exponential velocity defect law is observed, was found to be very different from that derived under the assumption of parallel flow. It is found to be plausible that this indeed can be attributed to non-parallel effects. A small modification of the innermost part of the log-layer in the form of an additive constant within the log-function is predicted by the Lie group symmetry method. A qualitative agreement with such a behaviour just below the overlap region was found. The derived scaling law behaviour in the overlap region for the two-point correlation functions was also verified by the experimental data.

Analysis of velocity profile measurements from wind-tunnel experiments with saltation

Investigations of wind-field modification due to the presence of saltating sediments have relied heavily on wind tunnels, which are known to impose geometric constraints on full boundary layer development. There remains great uncertainty as to which portion of the vertical wind-speed profile to analyze when deriving estimates of shear velocity or surface roughness length because the lower sections are modified to varying degree by saltation, whereas the upper segments may be altered by artificially induced wake-like effects. Thus, it is not obvious which of several alternative velocity-profile parameterizations (e.g., Law of the Wall, Velocity Defect Law, Wake Law) should be employed under such circumstances. A series of experimental wind-tunnel runs was conducted across a range of wind speed using fine- and coarse-grained sand to collect high-quality, fine-resolution data within and above the saltation layer using thermal anemometry and ruggedized probes. After each run, the rippled bottom was fixed with fine mist, and the experiment repeated without saltation. The measured wind-speed profiles were analyzed using six different approaches to derive estimates of shear velocity and roughness length. The results were compared to parameter estimates derived directly from sediment transport rate measurements, and on this basis, it is suggested that one of the six approaches is more robust than the others. Specifically, the best estimate of shear velocity during saltation is provided by the logarithmic law applied to the profile data within about 0.05 m of the bottom, despite the fact that this near-surface region is where profile modification by saltating sediments is most pronounced. Uncertainty remains as to whether this conclusion can be generalized to field situations because progressive downwind adjustments in the interrelationship between the saltation layer and the wind field are anticipated in wind tunnels, thereby confounding most analyses based on equilibrium assumptions.

WS3-4 カルマン定数と実験流体力学

Local similarity laws such as log-law and velocity defect law in wall turbulence are discussed from the point of view of Experimental Fluid Dynamics (EFD). Emphasis is located on numerical constants involved in the similarity laws, that is, Karman constant which value is indeed important in engineering applications. It is decided that only experiments can give last for the issue. Approach to the similarity law with observation of large-scale eddies is demonstrated by utilizing transport equation for two-point correlation function.

Scaling Law for Couette-Poiseuille Type Turbulent Flow and Study on 1/2 Power Law and Core Regions.

Direct numerical simulations of fully developed Couette-Poiseuille type turbulent flow are performed. The velocity law of this flow depends on the Reynolds number and two non-dimensional parameters of the shear stress gradient. We pay a particular attention to the effect of the shear stress gradient on the velocity distribution on the Couette-type flow. Computational results are dependent on the three non-dimensional parameters and are compared with theoretical laws and existing experimental data. The law of the mean velocity profile for Couette-Poiseuille type turbulent flow is considered in the regions characterized by 1/2-power law and velocity defect law. In particular, it is shown that the influence of turbulent diffusion from the opposite wall appears in the flow with large shear stress gradient and 1/2-power law is restricted to narrow region.

A possible reinterpretation of the Princeton superpipe data

In experiments recently performed at Melbourne, Pitot-tube mean velocity profiles in a boundary layer disagreed with those obtained with hot wires. The standard MacMillan (1956) correction for the probe displacement effect and a correction for turbulence intensity were both required for obtaining agreement between the two sets of mean velocity data. We were thus motivated to reanalyse the Princeton superpipe data using the same two corrections. The result is a plausible conclusion that the superpipe is rough at the higher Reynolds numbers and its data follow the Colebrook (1939) formula for commercial pipes rather well. It also appears that the logarithmic law of the wall is valid, with a Kármán constant close to that found recently by Österlund (1999) from boundary layer measurements with a hot wire. The smooth regime in the pipe gave almost the same additive constant for the log-law as Österlund's. A comparison between the superpipe data and the pipe data of Perry, Henbest & Chong (1997) suggests that the conventional velocity defect law may be valid down to lower Reynolds numbers than concluded by Zagarola & Smits (1998).

The Flow over an Inclined Fence in a Turbulent Boundary Layer.

The purpose of this paper is to study in detail the flow over an inclined fence in a turbulent boundary layer, as basic research which issues from promoting heat transfer and the decrease of the pressure drop. The fence height h set in a turbulent boundary layer is 3 mm (uτh/ν≒75, uτ : friction velocity, ν : kinematic viscosity) and setting angle of a fence is 20° in the free stream direction. The time averaged velocity profiles in the downstream of the fence ware measured by using a 3-hole pitot tube. We investigated velocity profiles by utilizing polar diagram, logarithmic velocity distributions and velocity defect law. Consequently, It was found that the spanwise velocity profiles are cross-over profiles. We clarified the relaxation process from streamwise direction velocity. Further we investigated in detail the flow direction on the wall which shows the wall shear direction by the oil flow visualization.

Resuspension and Redistribution of Sediments in Hamilton Harbour

The size distribution of sediments in Hamilton Harbour is a result of both natural and human processes. The objective is to predict the behavior of sediment particles in the harbor, assuming wind driven resuspension to be the major natural factor controlling grain size distribution. In this approach hydraulic sorting of sediments was assumed to be entirely predictable from 10 years of wind direction and velocity data used to calculate fetch over the extent of the harbor. Bottom velocities were estimated assuming a universal velocity defect law and the direction of bottom currents were assumed to parallel the depth contours. Although simplistic in many regards, the model returns adjustment parameters that are consistent with accepted values and is sufficiently articulated to be used as a reference to evaluate the cause of sediment transport in the harbor. Known local disturbances associated with human activity in the harbor could be predicted by mapping the differences between sediment distribution and actual core data from surficial sediments. A large part of the sediment in the deeper part of the harbor however contained more silt and clays than predicted. We assume this excess to be caused by a uniform disturbance equivalent to a 4.8 cm/sec bottom current and results from the combination of small bottom currents and other transfers of energy from the mixed layer. Historical records show that this situation existed long before the industrial period. Man-made disturbances on the overall distribution of bottom sediments are therefore less important compared to changes due to agricultural practices at the turn of the century.

Asymptotic velocity defect profile in an incipient-separating axisymmetric flow

Presence of a new asymptotic mean velocity profile (velocity defect law) was found concomitant with appreciable incipient separation (instantaneous detachment > 20 % of the time) and extremely high-turbulence intensities in the final stages of an 8-deg total divergence angle conical diffuser fed with a fully developed turbulent pipe flow. Pulsed-wire anemometry was used for the measurement of incipient flow separation, mean velocities and skin-friction distributions. The conical diffuser flow shows some attributes .of quasiequilibrium in the final onethird length. In the second-half of the diffuser the mean velocity profiles indicate a distinct point of inflection, the wall regions have less sensitivity to the initial conditions, and the flow exhibits some insensitivity toward the local boundary conditions. A comparison with the other asymptotic velocity defect profiles in Perry-Schofie ld coordinates revealed the distinctly different shape of the present asymptotic profile found in the conical diffuser. The validity of this asymptotic mean velocity profile was confirmed at two Reynolds numbers. Application of rapid distortion theory indicated coexistence of the regions of slowly and rapidly changing turbulence in the conical diffuser. Results of the present study also support the theory that turbulence is not universal and depends on the initial and boundary conditions.

Experimental Study on Wavy Channel Turbulent Flow.

Measurements of a turbulent flow in a wavy channel of high aspect ratio, one wall being sinusoidal and the other flat, are described in this report. Attention is focused on the flow in the near field of the flat wall. Mean flow and major statistical quantities of turbulence such as turbulence intensity. Reynolds stresses and skewness factors are included. The near-wall region was influenced largely by periodical pressure variation and the viscous sublayer was thickened. In the mean time, the core region was little affected and the velocity defect law was little modified. As for the turbulence property, it was revealed that acceleration of flow vitalized the turbulence in the layer very close to the wall and deceleration attenuated it. These disturbances were reflected quickly in the turbulence intensity and turbulent energy production, but slowly in the shear stress. However, it turned out that the effect of the pressure gradient on the turbulence behavior could be explained in terms of flow structure.

Mean Flow Quantities for the Turbulent Boundary Layer over an Equivalent Smooth-Mesh Rough Wall.

Mean flow quantities have been investigated experimentally for the turbulent boundary layer developing over an equivalent smooth-mesh rough wall. The roughness Reynolds number defined as ratio of the roughness height to the viscous length is about 4 which corresponds to the fluid dynamical smoothness. Measurements were made within Reynolds number range of about 2200 to 6000. Mean velocity profile nondimensionalized by outer variables shows a similarity within all Reynolds numbers tested, but the present mean velocity profile increases in the inner layer compared with the smooth wall flow. The value of the local skill friction coefficient evaluated using five different methods was slightly lower than the value for smooth wall flow. For the present mesh wall flow, both logarithmic velocity and velocity defect laws are applicable, and the inclination of logarithmic linear part, namely Karman constant, is the same as for smooth wall boundary layer.

Rational extension of the Clauser eddy viscosity model to compressible boundary-layer flow

An extension of the Clauser eddy viscosity model for the outer part of a compressible, turbulent boundary layer is proposed. This extension, unlike all of the previous ones, is based on a derivation, and not on ad hoc assumptions. The logic of the derivation of the original model is followed, except that the Crocco integral and the Howarth-Dorodnitsyn transformation are used for the purpose of the comparison with the velocity defect law. Predictions with the new model and two earlier ad hoc extensions of the Clauser model are compared to measured data. Improved predictions are obtained with the new model for both the outer part of the layer and the skin friction coefficient.

An Experimental Study on the Transport of Sediment in Sewer Pipes with a Permanent Deposit

An experimental study of the transport of sediment in a part-full pipe was carried out in a concrete pipe. The experiments were confined to bedload transport. The purpose of this study was to analyze the flow conditions that characterize the stream traction in pipe channels and their relationship to flow resistance and sediment transport rate. Three procedures used in this kind of experimental study were tested and found valid: 1) the vertical velocity distribution near the sediment bed can be described by the velocity-defect law, 2) the side wall elimination procedure can be used to compute the hydraulic radius of the sediment bed, and 3) the critical shear stress of the sediment particles can be obtained by using Shields' diagram. A relationship to estimate bedload transport, based on dimensional analysis, was proposed. This was expressed in terms of both flow and particle parameters as well as geometric factors. Further experimental work is recommended before this relationship can be fully incorporated in a simulation model for the analysis of storm sewers.

Turbulent Couette flow profiles that maximize the efficiency function

A previous study revealed that only maximization of the efficiency function, from among a large class of mean field moments, results in both a logarithmic law and a velocity defect law in turbulent Poiseuille channel flow. The efficiency function, [Escr ], is the product of a drag coefficient and the ratio of the fluctuation and mean dissipation rate integrals. Here, maximum [Escr ] is explored in Couette flow to test its generality as a statistical stability criterion for turbulent shear flows. The optimal flow exhibits a logarithmic law but does not have a velocity defect law. A decreasing velocity defect is predicted for Reynolds numbers up to 30 000. This prediction is shown to be supported by the existing data, which are limited to Reynolds numbers less than 20 000.